def zeta_symmetric(s):
    r"""
    Completed function `\xi(s)` that satisfies
    `\xi(s) = \xi(1-s)` and has zeros at the same points as the
    Riemann zeta function.

    INPUT:


    -  ``s`` - real or complex number


    If s is a real number the computation is done using the MPFR
    library. When the input is not real, the computation is done using
    the PARI C library.

    More precisely,

    .. math::

                xi(s) = \gamma(s/2 + 1) * (s-1) * \pi^{-s/2} * \zeta(s).



    EXAMPLES::

        sage: zeta_symmetric(0.7)
        0.497580414651127
        sage: zeta_symmetric(1-0.7)
        0.497580414651127
        sage: RR = RealField(200)
        sage: zeta_symmetric(RR(0.7))
        0.49758041465112690357779107525638385212657443284080589766062
        sage: C.<i> = ComplexField()
        sage: zeta_symmetric(0.5 + i*14.0)
        0.000201294444235258 + 1.49077798716757e-19*I
        sage: zeta_symmetric(0.5 + i*14.1)
        0.0000489893483255687 + 4.40457132572236e-20*I
        sage: zeta_symmetric(0.5 + i*14.2)
        -0.0000868931282620101 + 7.11507675693612e-20*I

    REFERENCE:

    - I copied the definition of xi from
      http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html
    """
    if not (is_ComplexNumber(s) or is_RealNumber(s)):
        s = ComplexField()(s)

    R = s.parent()
    if s == 1:  # deal with poles, hopefully
        return R(0.5)

    return (s / 2 + 1).gamma() * (s - 1) * (R.pi() ** (-s / 2)) * s.zeta()

    def map_to_complex_numbers(self, z, prec=None):
        """
        Evaluate ``self`` at a point `z \in X_0(N)` where `z` is given by
        a representative in the upper half plane, returning a point in
        the complex numbers.

        All computations are done with ``prec`` bits
        of precision.  If ``prec`` is not given, use the precision of `z`.
        Use self(z) to compute the image of z on the Weierstrass equation
        of the curve.

        EXAMPLES::

            sage: E = EllipticCurve('37a'); phi = E.modular_parametrization()
            sage: tau = (sqrt(7)*I - 17)/74
            sage: z = phi.map_to_complex_numbers(tau); z
            0.929592715285395 - 1.22569469099340*I
            sage: E.elliptic_exponential(z)
            (...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000)
            sage: phi(tau)
            (...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000)
        """
        if prec is None:
            try:
                prec = z.parent().prec()
            except AttributeError:
                prec = 53
        CC = ComplexField(prec)
        if z in QQ:
            raise NotImplementedError
        z = CC(z)
        if z.imag() <= 0:
            raise ValueError("Point must be in the upper half plane")
        # TODO: for very small imaginary part, maybe try to transform under
        # \Gamma_0(N) to a better representative?
        q = (2 * CC.gen() * CC.pi() * z).exp()
        #  nterms'th term is less than 2**-(prec+10) (c.f. eclib code)
        nterms = (-(prec + 10) / q.abs().log2()).ceil()
        # Use Horner's rule to sum the integral of the form
        enumerated_an = list(enumerate(self._E.anlist(nterms)))[1:]
        lattice_point = 0
        for n, an in reversed(enumerated_an):
            lattice_point += an / n
            lattice_point *= q
        return lattice_point

    def _evalf_(self, x, y, parent=None, algorithm='mpmath'):
        """
        EXAMPLES::

            sage: gamma_inc_lower(3,2.)
            0.646647167633873
            sage: gamma_inc_lower(3,2).n(200)
            0.646647167633873081060005050275155...
            sage: gamma_inc_lower(0,2.)
            +infinity
        """
        R = parent or s_parent(x)
        # C is the complex version of R
        # prec is the precision of R
        if R is float:
            prec = 53
            C = complex
        else:
            try:
                prec = R.precision()
            except AttributeError:
                prec = 53
            try:
                C = R.complex_field()
            except AttributeError:
                C = R
        if algorithm == 'pari':
            try:
                v = ComplexField(prec)(x).gamma() - ComplexField(prec)(x).gamma_inc(y)
            except AttributeError:
                if not (is_ComplexNumber(x)):
                    if is_ComplexNumber(y):
                        C = y.parent()
                    else:
                        C = ComplexField()
                        x = C(x)
            v = ComplexField(prec)(x).gamma() - ComplexField(prec)(x).gamma_inc(y)
        else:
            import mpmath
            v = ComplexField(prec)(mpmath_utils.call(mpmath.gammainc, x, 0, y, parent=R))
        if v.is_real():
            return R(v)
        else:
            return C(v)

    def __init__(self, conductor, gammaV, weight, eps, \
                       poles=[], residues='automatic', prec=53,
                       init=None):
        """
        Initialization of Dokchitser calculator EXAMPLES::

            sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
            sage: L.num_coeffs()
            4
        """
        self.conductor = conductor
        self.gammaV = gammaV
        self.weight = weight
        self.eps = eps
        self.poles = poles
        self.residues = residues
        self.prec = prec
        self.__CC = ComplexField(self.prec)
        self.__RR = self.__CC._real_field()
        self.__initialized = False
        if init is not None:
            self.init_coeffs(init)

def hilbert_class_polynomial(D, algorithm=None):
    r"""
    Returns the Hilbert class polynomial for discriminant `D`.

    INPUT:

    - ``D`` (int) -- a negative integer congruent to 0 or 1 modulo 4.

    - ``algorithm`` (string, default None).

    OUTPUT:

    (integer polynomial) The Hilbert class polynomial for the
    discriminant `D`.

    ALGORITHM:

    - If ``algorithm`` = "arb" (default): Use Arb's implementation which uses complex interval arithmetic.

    - If ``algorithm`` = "sage": Use complex approximations to the roots.

    - If ``algorithm`` = "magma": Call the appropriate Magma function (if available).

    AUTHORS:

    - Sage implementation originally by Eduardo Ocampo Alvarez and
      AndreyTimofeev

    - Sage implementation corrected by John Cremona (using corrected precision bounds from Andreas Enge)

    - Magma implementation by David Kohel

    EXAMPLES::

        sage: hilbert_class_polynomial(-4)
        x - 1728
        sage: hilbert_class_polynomial(-7)
        x + 3375
        sage: hilbert_class_polynomial(-23)
        x^3 + 3491750*x^2 - 5151296875*x + 12771880859375
        sage: hilbert_class_polynomial(-37*4)
        x^2 - 39660183801072000*x - 7898242515936467904000000
        sage: hilbert_class_polynomial(-37*4, algorithm="magma") # optional - magma
        x^2 - 39660183801072000*x - 7898242515936467904000000
        sage: hilbert_class_polynomial(-163)
        x + 262537412640768000
        sage: hilbert_class_polynomial(-163, algorithm="sage")
        x + 262537412640768000
        sage: hilbert_class_polynomial(-163, algorithm="magma") # optional - magma
        x + 262537412640768000

    TESTS::

        sage: all([hilbert_class_polynomial(d, algorithm="arb") == \
        ....:      hilbert_class_polynomial(d, algorithm="sage") \
        ....:        for d in range(-1,-100,-1) if d%4 in [0,1]])
        True

    """
    if algorithm is None:
        algorithm = "arb"

    D = Integer(D)
    if D >= 0:
        raise ValueError("D (=%s) must be negative"%D)
    if not (D%4 in [0,1]):
         raise ValueError("D (=%s) must be a discriminant"%D)

    if algorithm == "arb":
        import sage.libs.arb.arith
        return sage.libs.arb.arith.hilbert_class_polynomial(D)

    if algorithm == "magma":
        magma.eval("R<x> := PolynomialRing(IntegerRing())")
        f = str(magma.eval("HilbertClassPolynomial(%s)"%D))
        return IntegerRing()['x'](f)

    if algorithm != "sage":
        raise ValueError("%s is not a valid algorithm"%algorithm)

    from sage.quadratic_forms.binary_qf import BinaryQF_reduced_representatives
    from sage.rings.all import RR, ZZ, ComplexField
    from sage.functions.all import elliptic_j

    # get all primitive reduced quadratic forms, (necessary to exclude
    # imprimitive forms when D is not a fundamental discriminant):

    rqf = BinaryQF_reduced_representatives(D, primitive_only=True)

    # compute needed precision
    #
    # NB: [http://arxiv.org/abs/0802.0979v1], quoting Enge (2006), is
    # incorrect.  Enge writes (2009-04-20 email to John Cremona) "The
    # source is my paper on class polynomials
    # [http://hal.inria.fr/inria-00001040] It was pointed out to me by
    # the referee after ANTS that the constant given there was
    # wrong. The final version contains a corrected constant on p.7
    # which is consistent with your example. It says:

    # "The logarithm of the absolute value of the coefficient in front
#.........这里部分代码省略.........

#.........这里部分代码省略.........
        12
        sage: L.set_coeff_growth('2*n^(11/2)')
        sage: L.num_coeffs()
        11

    Now we're ready to evaluate, etc.

    ::

        sage: L(1)
        0.0374412812685155
        sage: L(1, 1.1)
        0.0374412812685155
        sage: L.taylor_series(1,3)
        0.0374412812685155 + 0.0709221123619322*z + 0.0380744761270520*z^2 + O(z^3)
    """
    def __init__(self, conductor, gammaV, weight, eps, \
                       poles=[], residues='automatic', prec=53,
                       init=None):
        """
        Initialization of Dokchitser calculator EXAMPLES::

            sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
            sage: L.num_coeffs()
            4
        """
        self.conductor = conductor
        self.gammaV = gammaV
        self.weight = weight
        self.eps    = eps
        self.poles  = poles
        self.residues = residues
        self.prec   = prec
        self.__CC   = ComplexField(self.prec)
        self.__RR   = self.__CC._real_field()
        if not init is None:
            self.init_coeffs(init)
            self.__init = init
        else:
            self.__init = False

    def __reduce__(self):
        D = copy.copy(self.__dict__)
        if '_Dokchitser__gp' in D:
            del D['_Dokchitser__gp']
        return reduce_load_dokchitser, (D, )

    def _repr_(self):
        z = "Dokchitser L-series of conductor %s and weight %s"%(
                   self.conductor, self.weight)
        return z

    def __del__(self):
        self.gp().quit()

    def gp(self):
        """
        Return the gp interpreter that is used to implement this Dokchitser
        L-function.

        EXAMPLES::

            sage: E = EllipticCurve('11a')
            sage: L = E.lseries().dokchitser()
            sage: L(2)
            0.546048036215014