A Dirichlet charachter is a function (here is the non-zero elements mod and are the non-zero complex numbers) which satisfies .
Given a Dirichlet charachter we form the Dirichlet L-series where is complex.
It can be proven that converges when .
Using analytic continuation we extend this to a meromorphic function on .
This continued new function is called the Dirichlet L-function.
It is related to Riemann zeta function.