---
tag: 9D2Z543EJQ
kind: concept
name: Dirichlet L-function
msc: [11M06]
specializes: [8MWTEF6462]
sources:
  - url: https://en.wikipedia.org/wiki/Dirichlet_L-function
same_as:
  wikidata: Q2264419
  wikipedia: Dirichlet L-function
---

## Summary

A Dirichlet L-function is the [[Dirichlet series|8MWTEF6462]] $L(s, \chi) = \sum_{n \ge 1} \chi(n) n^{-s}$ of a [[Dirichlet character|7XB5VJKDPG]] $\chi$, together with its analytic continuation.
It has an Euler product over the primes and generalizes the [[Riemann zeta function|1XWBY4N0VT]], which is the case of the trivial character modulo $1$.
The fact that $L(1, \chi) \ne 0$ for every nonprincipal $\chi$ is the heart of [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]].

## Explanations

### L1 human

The zeta function adds $1/n^s$ over all positive integers.
A Dirichlet L-function adds the same terms with the weights $\chi(n)$, which depend on $n$ modulo $q$.
These weights let one separate the primes according to their remainder modulo $q$.
For the character modulo $4$ with values $1, 0, -1, 0$, the value at $s = 1$ is $1 - 1/3 + 1/5 - 1/7 + \cdots = \pi/4$.