Dirichlet L-function

Definition

A Dirichlet L-function is a meromorphic function obtained as the analytic continuation of the Dirichlet series of the Dirichlet character.

The corresponding Dirichlet series is termed a Dirichlet L-series, and the Dirichlet L-series as well as the function are denoted $L(s,\chi )$ where $\chi$ is the character and $s$ is the argument.

$L(s,\chi ):={\sum }_{n\in \mathbb{N}}\frac{\chi (n)}{n^s}={\prod }_{p\in \mathbb{P}}\frac{1}{1-\chi (p)p^{-s}}$,

where $\mathbb{P}$ is the set of all prime numbers. The equality follows from the Euler product formula.