Modulus of a Dirichlet character (reviewed)

A Dirichlet character is a function $\chi: \mathbb Z\to \mathbb C$ together with a positive integer $q$, called the modulus of the character, such that $\chi$ is completely multiplicative, i.e. $\chi(mn)=\chi(m)\chi(n)$ for all integers $m$ and $n$, and $\chi$ is periodic modulo $q$, i.e. $\chi(n+q)=\chi(n)$ for all $n$. If $(n,q)>1$ then $\chi(n)=0$, whereas if $(n,q)=1$, then $\chi(n)$ is a root of unity.