Dirichlet character

Dirichlet character (plural Dirichlet characters)

1. (analytic number theory) A complex-valued arithmetic function ${\displaystyle \chi$ :\mathbb {Z} \rightarrow \mathbb {C} } that satisfies the following conditions for some positive integer ${\displaystyle m}$ and all integers ${\displaystyle a}$ and ${\displaystyle b}$:

   1)   ${\displaystyle \chi (ab)=\chi (a)\chi (b);}$  

   2)   ${\displaystyle \chi (a){\begin{cases}=0&{\text{if }}\;\gcd(a,m)>1\\\neq 0&{\text{if }}\;\gcd(a,m)=1.\end{cases}}}$ (gcd is the greatest common divisor)

   3)   ${\displaystyle \chi (a+m)=\chi (a)}$.