# Definition:Dirichlet L-function

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## Definition

Let $\chi : \paren {\Z / q \Z}^\times \to \C^\times$ be a Dirichlet character.

A **Dirichlet $L$-function (associated to $\chi$)** is a Dirichlet series:

- $\ds \map L {s, \chi} = \sum_{n \mathop \ge 1} \map \chi n n^{-s}$

for all $s \in \C$ such that the sum converges.

## Also see

This is extended to the complex plane by Analytic Continuation of Dirichlet L-functions.

## Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.