I'm trying to decipher Lang's Algebraic Number Theory when it comes to L-functions. Let $K/\mathbb{Q}$ be an abelian extension. Then we are supposed to have a factorization
$$\zeta_K(s)=\zeta_\mathbb{Q}(s)\prod_{\chi\neq 1} L(s,\chi)$$
I'm trying to understand this factorization in more concrete terms. First of all Lang gives the definition of $\chi$ as a group on ideles. I was wondering if there is a simple more concrete and computable definition in terms of ray classes?
Let's say we take some simple example like $\mathbb{Q}(\sqrt{-3})$. In what group are these $\chi$ defined and how would I compute them? Is there some algorithm that works in general? I understand that this has something to do with the conductor, which would be $(3)p_\infty$ in the above case.