# Artin L-function

This term is related to: Galois theory
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## Definition

Let $L/K$ be a Galois extension of fields, and $G$ its Galois group. Let $\rho$be a linear representation of $G$ over $\mathbb{C}$. (In other words, $\rho$ is a Galois representation over the complex numbers).

The Artin L-function associated with $\rho$, denoted as $s \mapsto L(\rho,s)$, is defined as follows: it is the product, over all prime ideals $P$, of the following Euler factor corresponding to that $P$:

$det(1 - t\rho(Frob(P)))^{-1}$

evaluated at $t = N(P^{-s})$.

Strictly speaking, the above definition works when $P$ is unramified. A slight variant works when $P$ is ramified.

When the underlying Galois group $G$ is Abelian, the Artin L-function specializes to the Hecke L-function.
When $G$ is Abelian and $K = \mathbb{Q}$, the Artin L-function specializes to the Dirichlet L-function.