Artin L-function

This term is related to: Galois theory
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This article defines a type of L-function

This article or section of article is sourced from:Wikipedia

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.

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Particular cases

For Abelian Galois group

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

For Abelian Galois group and over rationals

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

External links

Wikipedia page