Find examples (the more the better) of functions $f: \mathbb{Z} \to \mathbb{C}$ satisfying the relations

1. $f(x+y) = f(x) + f(y)$
2. $f(xy) = f(x)f(y)$

I have only

1. $f(x)= ax$
2. $f(x)= x^a$

This task is from algebra but I don't know why.