Dirichlet character

Noun
\chi(a) \begin{cases} =0 &\text{if }\; \gcd(a,m)>1\\ \ne 0&\text{if }\;\gcd(a,m)=1. \end{cases}$$ (gcd is the greatest common divisor)
 * 1)  A complex-valued arithmetic function  $$\chi:\mathbb{Z}\rightarrow\mathbb{C}$$ that satisfies the following conditions for some positive integer $$m$$ and all integers $$a$$ and $$b$$:
 * 1)  $$\chi(ab) = \chi(a)\chi(b);$$
 * 2)  $$
 * 3)  $$\chi(a + m) = \chi(a)$$.