lfuntheta

Function: lfuntheta
Section: l_functions
C-Name: lfuntheta
Prototype: GGD0,L,b
Help: lfuntheta(data,t,{m=0}): compute the value of the m-th derivative
 at t of the theta function attached to the L-function given by data.
 data can be either the standard L-function data, or the output of
 lfunthetainit.
Doc: compute the value of the $m$-th derivative
 at $t$ of the theta function attached to the $L$-function given by \kbd{data}.
  \kbd{data} can be either the standard $L$-function data, or the output of
 \kbd{lfunthetainit}. The result is given with absolute error less than
 $2^{-B}$, where $B = \text{realbitprecision}$.

 The theta function is defined by the formula
 $\Theta(t)=\sum_{n\ge1}a(n)K(nt/\sqrt(N))$, where $a(n)$ are the coefficients
 of the Dirichlet series, $N$ is the conductor, and $K$ is the inverse Mellin
 transform of the gamma product defined by the \kbd{Vga} component.
 Its Mellin transform is equal to $\Lambda(s)-P(s)$, where $\Lambda(s)$
 is the completed $L$-function and the rational function $P(s)$ its polar part.
 In particular, if the $L$-function is the $L$-function of a modular form
 $f(\tau)=\sum_{n\ge0}a(n)q^n$ with $q=\exp(2\pi i\tau)$, we have
 $\Theta(t)=2(f(it/\sqrt{N})-a(0))$. Note that $a(0)=-L(f,0)$ in this case.