# Definition:Dirichlet L-function

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## Definition

Let $\chi : \left({\Z / q \Z}\right)^\times \to \C^\times$ be a Dirichlet character.

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

- $\displaystyle L \left({s, \chi}\right) = \sum_{n \mathop \ge 1} \chi \left({n}\right) n^{-s}$

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.