# Artin L-function

This term is related to: Galois theory

View other terms related to Galois theory | View facts related to Galois theory

*This article defines a type of L-function*

*This article or section of article is sourced from*:Wikipedia

## Contents

## Definition

Let be a Galois extension of fields, and its Galois group. Let be a linear representation of over . (In other words, is a Galois representation over the complex numbers).

The Artin L-function associated with , denoted as , is defined as follows: it is the product, over all prime ideals , of the following Euler factor corresponding to that :

evaluated at .

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

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## Particular cases

### For Abelian Galois group

When the underlying Galois group is Abelian, the Artin L-function specializes to the Hecke L-function.

### For Abelian Galois group and over rationals

When is Abelian and , the Artin L-function specializes to the Dirichlet L-function.