Class LCalc

Return the imaginary parts of the first $n$ nontrivial zeros
of the $L$-function in the upper half plane, as 32-bit reals.

INPUT:
    n -- integer
    L -- defines $L$-function (default: Riemann zeta function)

This function also checks the Riemann Hypothesis and makes
sure no zeros are missed.  This means it looks for several
dozen zeros to make sure none have been missed before
outputting any zeros at all, so takes longer than
\code{self.zeros_of_zeta_in_interval(...)}.

EXAMPLES:
    sage: lcalc.zeros(4)                           # long
    [14.1347251, 21.0220396, 25.0108576, 30.4248761]
    sage: lcalc.zeros(5, L='--tau')                # long
    [9.22237940, 13.9075499, 17.4427770, 19.6565131, 22.3361036]
    sage: lcalc.zeros(3, EllipticCurve('37a'))     # long
    [0.000000000, 5.00317001, 6.87039122]

Return the imaginary parts of (most of) the nontrivial zeros
of the $L$-function on the line $\Re(s)=1/2$ with
positive imaginary part between $x$ and $y$, along with a
technical quantity for each.

INPUT:
    x, y, stepsize -- positive floating point numbers
    L -- defines $L$-function (default: Riemann zeta function)

OUTPUT:
    list of pairs (zero, S(T)).

Rubinstein writes: The first column outputs the imaginary part
of the zero, the second column a quantity related to $S(T)$ (it
increases roughly by 2 whenever a sign change, i.e. pair of
zeros, is missed). Higher up the critical strip you should use
a smaller stepsize so as not to miss zeros.

EXAMPLES:
    sage: lcalc.zeros_in_interval(10, 30, 0.1)
    [(14.1347251, 0.184672916), (21.0220396, -0.0677893290), (25.0108576, -0.0555872781)]

Return $L(s)$ for $s$ a complex number.

INPUT:
    s -- complex number
    L -- defines $L$-function (default: Riemann zeta function)

EXAMPLES:
    sage: I = CC.0
    sage: lcalc.value(0.5 + 100*I)
    2.69261989 - 0.0203860296*I

Note, SAGE can also compute zeta at complex numbers (using
the PARI C library):
    sage: (0.5 + 100*I).zeta()
    2.69261988568132 - 0.0203860296025982*I

Return values of $L(s)$ at \code{number_samples}
equally-spaced sample points along the line from $s_0$ to
$s_1$ in the complex plane.

INPUT:
    s0, s1 -- complex numbers
    number_samples -- integer
    L -- defines $L$-function (default: Riemann zeta function)
    
OUTPUT:
    list -- list of pairs (s, zeta(s)), where the s are
            equally spaced sampled points on the line from
            s0 to s1.

EXAMPLES:
    sage: I = CC.0
    sage: lcalc.values_along_line(0.5, 0.5+20*I, 5)
    [(0.500000000, -1.46035451), (0.500000000 + 4.00000000*I, 0.606783764 + 0.0911121400*I), (0.500000000 + 8.00000000*I, 1.24161511 + 0.360047588*I), (0.500000000 + 12.0000000*I, 1.01593665 - 0.745112472*I), (0.500000000 + 16.0000000*I, 0.938545408 + 1.21658782*I)]

Sometimes warnings are printed (by lcalc) when this command is run:
    sage: E = EllipticCurve('389a')
    sage: E.lseries().values_along_line(0.5, 3, 5)
    [(0, 0.209951303),
     (0.500000000, -...e-16),
     (1.00000000, 0.133768433),
     (1.50000000, 0.360092864),
     (2.00000000, 0.552975867)]

Return values of $L(s, \chi_d)$ for each quadratic
character $\chi_d$ for $d_{\min} \leq d \leq d_{\max}$.

INPUT:
    s -- complex numbers
    dmin -- integer
    dmax -- integer
    L -- defines $L$-function (default: Riemann zeta function)

OUTPUT:
    list -- list of pairs (d, L(s,chi_d))

EXAMPLES:
    sage: lcalc.twist_values(0.5, -10, 10)
    [(-8, 1.10042141), (-7, 1.14658567), (-4, 0.667691457), (-3, 0.480867558), (5, 0.231750947), (8, 0.373691713)]

Return first $n$ real parts of nontrivial zeros for each
quadratic character $\chi_d$ for $d_{\min} \leq d \leq
d_{\max}$.

INPUT:
    n -- integer
    dmin -- integer
    dmax -- integer
    L -- defines $L$-function (default: Riemann zeta function)

OUTPUT:
    dict -- keys are the discriminants $d$, and
            values are list of corresponding zeros.

EXAMPLES:
    sage: lcalc.twist_zeros(3, -3, 6)
    {-3: [8.03973716, 11.2492062, 15.7046192], 5: [6.64845335, 9.83144443, 11.9588456]}

Return the analytic rank of the $L$-function at the central
critical point.

INPUT:
    L -- defines $L$-function (default: Riemann zeta function)

OUTPUT:
    integer
    
\note{Of course this is not provably correct in general, since
it is an open problem to compute analytic ranks provably
correctly in general.}

EXAMPLES:
    sage: E = EllipticCurve('37a')
    sage: lcalc.analytic_rank(E)
    1