Artin L-function

An Artin L-function encodes traces of Frobenius elements in a Galois representation and converts character sums over primes into analytic information.

Only the inertia group invariants enter an Artin L-function local factor. A Frobenius automorphism lift acts unambiguously on this subspace, since other lifts differ by inertia. For an unramified one-dimensional group representation, the factor is $(1-\chi ({\mathrm{Frob}}_{\mathfrak{p}})(N\mathfrak{p}{)}^{-s}{)}^{-1}$; for a ramified one-dimensional group representation, the invariant subspace is zero and the factor is one. This explains the zero values used at conductor primes by quadratic Dirichlet characters.

For the standard representation of the symmetric group $S_3$, the unramified local factor of an Artin L-function, with $t=p^{-s}$, is respectively the displayed factor for identity, transposition, and three-cycle Frobenius cycle types. These follow by removing the trivial summand from the three-point permutation representation. If the inertia group contains a three-cycle, its invariant subspace in the standard representation is zero, so the ramified local factor is one.

Assume that every nontrivial one-dimensional Artin L-function over the intermediate fields is holomorphic and nonzero at one, while the trivial one is the base Dedekind zeta function with a simple pole. By Brauer induction, $\chi ={\sum }_j{n}_j{\mathrm{Ind}}_{H_j}^G{\psi }_j$ with integers $n_j$ and linear characters ${\psi }_j$. Compatibility of Artin L-functions with induced representations expresses $L_K(\chi ,s)$ as the corresponding product of $L_{L^{H_j}}({\psi }_j,s{)}^{n_j}$. Its order at one is $-{\sum }_j{n}_j⟨{\psi }_j,1_{H_j}⟩$. Frobenius reciprocity turns this sum into $-⟨\chi ,1_G⟩$. Hence every nontrivial irreducible character gives order zero and a holomorphic nonzero function near one. This local conclusion does not assert global Artin holomorphy.