TITLE: Factorization of L-functions
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: In my mind, the concrete example of this factorization is for the Gaussian integers. In this case we have that $$\chi(n) =
\begin{cases}
1 \;\;\;\;\;\; \text{if } n \equiv 1 \mod{4} \\
-1 \;\;\;\; \text{if } n \equiv 3 \mod{4}\\
0 \; \; \; \; \; \text{otherwise}
\end{cases}$$
is a Dirichlet character $\chi$ of $\mathbb{Z}[i]$, $L(s,\chi)$ is the Dirichlet L-function evaluated at $\chi$, and $\zeta_{\mathbb{Q}(i)}(s)$ is the Dedekind zeta function over the Gaussian integers. Then,
$$\zeta_{\mathbb{Q}(i)}(s) = \zeta(s)L(s,\chi)$$
is the factorization of the Dedekind zeta function as a product of the Riemann zeta function and a Dirichlet L-function over $\mathbb{Z}[i]$.