# Dirichlet character

In number theory, a Dirichlet character is a multiplicative function on the positive integers which is derived from a character on the multiplicative group taken modulo a given integer.

Let *N* be a positive integer and write (**Z**/*N*)* for the multiplicative group of integers modulo *N*. Let χ be a group homomorphism from (**Z**/*N*)* to the unit circle. Since the multiplicative group is finite of order φ(*N*), where φ is the Euler totient function, the values of χ are all roots of unity. We extend χ to a function on the positive integers by defining χ(*n*) to be χ(*n* mod *N*) when *n* is coprime to *N*, and to be zero when *n* has a factor in common with *N*. This extended function is the *Dirichlet character*. As a function on the positive integers it is a totally multiplicative function with period *n*.

The *principal character* χ_{0} is derived from the trivial character which is 1 one *n* coprime to *N* and zero otherwise.

We say that a Dirichlet character χ_{1} with modulus *N*_{1} "induces" χ with modulus *N* if *N*_{1} divides *N* and χ(*n*) agrees with χ_{1}(*n*) whenever they are both non-zero. A *primitive* character is one which is not induced from any character with smaller modulus. The *conductor* of a character is the modulus of the associated primitive character.

## Dirichlet L-function

The **Dirichlet L-function** associated to χ is the Dirichlet series

with an Euler product

If χ is principal then *L*(*s*,χ) is the Riemann zeta function with finitely many Euler factors removed, and hence has a pole of order 1 at *s*=1. Otherwise *L*(*s*,χ) has a half-plane of convergence to the right of *s*=0. In all cases, *L*(*s*,χ) has an analytic continuation to the complex plane with a functional equation.