Function: lfunartin
Section: l_functions
C-Name: lfunartin
Prototype: GGGL
Help: lfunartin(nf,gal,M,n): returns the Ldata structure attached to the
 Artin L-function attached to the representation R of the Galois group
 of the extension K/Q, defined over the cyclotomic field Q(zeta_n),
 where nf is the nfinit structure attached to K,
 gal is the galoisinit structure attached to K/Q, and M is the vector
 of the image of the generators G.gen by R. The elements of M are matrices
 with polynomial entries, whose variable is understood as the complex
 number exp(2*I*Pi/n).
Doc: returns the \kbd{Ldata} structure attached to the
 Artin $L$-function attached to the representation $\rho$ of the Galois group
 of the extension $K/\Q$, defined over the cyclotomic field $\Q(\zeta_n)$,
 where \var{nf} is the nfinit structure attached to $K$,
 \var{gal} is the galoisinit structure attached to $K/\Q$, and $M$ is
 the vector of the image of the generators \kbd{\var{gal}.gen} by $\rho$.
 The elements of $M$ are matrices with polynomial entries, whose variable
 is understood as the complex number $\exp(2\*i\*\pi/n)$.

 In the following example we build the Artin $L$-functions attached to the two
 irreducible degree $2$ representations of the dihedral group $D_{10}$ defined
 over $\Q(\zeta_5)$, for the extension $H/\Q$ where $H$ is the Hilbert class
 field of $\Q(\sqrt{-47})$.
 We show numerically some identities involving Dedekind $\zeta$ functions and
 Hecke $L$ series.
 \bprog
 ? P = quadhilbert(-47);
 ? N = nfinit(nfsplitting(P));
 ? G = galoisinit(N);
 ? L1 = lfunartin(N,G, [[a,0;0,a^-1],[0,1;1,0]], 5);
 ? L2 = lfunartin(N,G, [[a^2,0;0,a^-2],[0,1;1,0]], 5);
 ? s = 1 + x + O(x^4);
 ? lfun(1,s)*lfun(-47,s)*lfun(L1,s)^2*lfun(L2,s)^2 - lfun(N,s)
 %6 ~ 0
 ? lfun(1,s)*lfun(L1,s)*lfun(L2,s) - lfun(P,s)
 %7 ~ 0
 ? bnr = bnrinit(bnfinit(x^2+47),1,1);
 ? lfun([bnr,[1]], s) - lfun(L1, s)
 %9 ~ 0
 ? lfun([bnr,[1]], s) - lfun(L1, s)
 %10 ~ 0
 @eprog
 The first identity is the factorisation of the regular representation of
 $D_{10}$, the second the factorisation of the natural representation of
 $D_{10}\subset S_5$, the next two are the expressions of the degree $2$
 representations as induced from degree $1$ representations.
