Artin L-function

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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 \rhobe 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