lfunrootres

Function: lfunrootres
Section: l_functions
C-Name: lfunrootres
Prototype: Gb
Help: lfunrootres(data): given the Ldata attached to an L-function (or the
 output of lfunthetainit), compute the root number and the
 residues. In the present implementation, if the polar part is not already
 known completely, at most a single pole is allowed.
 The output is a 3-component vector
  [[[a_1, r_1],...,[a_n, r_n],[[b_1, R_1],...[b_m,R_m]]~, w], where r_i is the
  polar part of L(s) at a_i, R_i is is the polar part of Lambda(s) at b_i,
  or [0,0,r] if there is no pole, and w is the root number.
Doc: Given the \kbd{Ldata} attached to an $L$-function (or the output of
 \kbd{lfunthetainit}), compute the root number and the residues.

 The output is a 3-component vector
 $[[[a_1,r_1],\cdots,[a_n, r_n], [[b_1, R_1],\cdots,[b_m, R_m]]~, w]$,
 where $r_i$ is the
 polar part of $L(s)$ at $a_i$, $R_i$ is is the polar part of $\Lambda(s)$ at
 $b_i$ or $[0,0,r]$ if there is no pole,
 and $w$ is the root number. In the present implementation,

 \item either the polar part must be completely known (and is then arbitrary):
 the function determines the root number,

 \bprog
 ? L = lfunmul(1,1); \\ zeta^2
 ? [r,R,w] = lfunrootres(L);
 ? r  \\ single pole at 1, double
 %3 = [[1, 1.[...]*x^-2 + 1.1544[...]*x^-1 + O(x^0)]]
 ? w
 %4 = 1
 ? R \\ double pole at 0 and 1
 %5 = [[1,[...]], [0,[...]]]~
 @eprog

 \item or at most a single pole is allowed: the function computes both
 the root number and the residue ($0$ if no pole).