 Artin L-function

For a finite Galois extension $L/K$ of number fields and a finite-dimensional complex group representation $\rho$ of its Galois group, the finite part of its Artin L-function is the Euler productHere $I_{\mathfrak{p}}$ is the inertia group, and a Frobenius automorphism lift acts on its invariant subspace. Different lifts induce the same operator there. At an unramified prime the invariant subspace is all of $V$; at a ramified prime this restriction removes the part on which inertia acts nontrivially.

The trivial representation gives the Dedekind zeta function of $K$. Direct sums of representations give products of Artin L-functions, because determinants multiply on direct sums. Taking the logarithm of each local factor produces traces of powers of Frobenius, connecting representation-theoretic characters with prime-counting sums. Archimedean factors are added separately when forming a completed L-function.

The Artin L-function is a generalization of the classical Riemann zeta function and is an important object in number theory and arithmetic geometry, particularly in the context of class field theory and algebraic number theory. It is associated with a representations of a Galois group, collections of characters, and the study of L-functions in the context of number fields. ### Definition 1.