\begin{definition}[Definition:Dirichlet L-function]
Let $\chi : \paren {\Z / q \Z}^\times \to \C^\times$ be a Dirichlet character.
{{explain|the notation $\paren {\Z / q \Z}^\times$}}
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.
\end{definition}