Function: lfunartin
Section: l_functions
C-Name: lfunartin
Prototype: GGGLb
Help: lfunartin(nf,gal,rho,n): returns the Ldata structure attached to the
 Artin L-function provided by the representation rho 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 rho is given either by the values of its character on the conjugacy
 classes or by the matrices that are the images of the generators. Cyclotomic
 numbers in rho are represented by polynomials, 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 provided by 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 \var{rho} is
 given either

 \item by the values of its character on the conjugacy classes
 (see \kbd{galoisconjclasses} and \kbd{galoischartable})

 \item or by the matrices that are the images of the generators
 \kbd{\var{gal}.gen}.

 Cyclotomic numbers in \kbd{rho} are represented by polynomials, 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)
 %1 = x^5 + 2*x^4 + 2*x^3 + x^2 - 1
 ? N = nfinit(nfsplitting(P));
 ? G = galoisinit(N); \\ D_10
 ? [T,n] = galoischartable(G);
 ? T  \\ columns give the irreducible characters
 %5 =
 [1  1              2              2]

 [1 -1              0              0]

 [1  1 -y^3 - y^2 - 1      y^3 + y^2]

 [1  1      y^3 + y^2 -y^3 - y^2 - 1]
 ? n
 %6 = 5
 ? L2 = lfunartin(N,G, T[,2], n);
 ? L3 = lfunartin(N,G, T[,3], n);
 ? L4 = lfunartin(N,G, T[,4], n);
 ? s = 1 + x + O(x^4);
 ? lfun(-47,s) - lfun(L2,s)
 %11 ~ 0
 ? lfun(1,s)*lfun(-47,s)*lfun(L3,s)^2*lfun(L4,s)^2 - lfun(N,s)
 %12 ~ 0
 ? lfun(1,s)*lfun(L3,s)*lfun(L4,s) - lfun(P,s)
 %13 ~ 0
 ? bnr = bnrinit(bnfinit(x^2+47),1,1);
 ? bnr.cyc
 %15 = [5] \\ Z/5Z: 4 non-trivial ray class characters
 ? lfun([bnr,[1]], s) - lfun(L3, s)
 %16 ~ 0
 ? lfun([bnr,[2]], s) - lfun(L4, s)
 %17 ~ 0
 ? lfun([bnr,[3]], s) - lfun(L3, s)
 %18 ~ 0
 ? lfun([bnr,[4]], s) - lfun(L4, s)
 %19 ~ 0
 @eprog
 The first identity identifies the non-trivial abelian character with
 $(-47,\cdot)$; the second is the factorization of the regular representation of
 $D_{10}$; the third is the factorization of the natural representation of
 $D_{10}\subset S_5$; and the final four are the expressions of the degree $2$
 representations as induced from degree $1$ representations.