What do we know about the set of Primitive Dirichlet Characters modulu an integer $N$?

Question

I was working on Dirichlet's theorem on arithmetic progressions and I have the following question:

Is the set of all Primitive Dirichlet Characters modulus some given (and fixed ) integer $N$ a ring for some known operations?

I tried some operations like the product $\chi_1\chi_2$ or the convolution $\chi_1*\chi_2$, but I did not have any important results