TITLE: Relationship between different L-functions
QUESTION [10 upvotes]: What's the relationship between between Artin $L$-functions and Dirichlet or Hecke $L$-functions if $L/K$ is an abelian extension? I've been told that one can interpret the Artin $L$-functions as these other $L$-functions through class field theory. Can anyone explain how this is done?
EDIT: I'll add this to make the harder part of the question more explicit. Given an Artin $L$-function $L(L/K,\rho,s)$ with $L/K$ abelian, how can one find a Hecke character $\chi$ s.t. $L(L/K,\rho,s)=L(s,\chi)$? I'm interested in seeing how this can be done explicitly through global CFT. I'm guessing the reciprocity law should make this somewhat "natural" given that it connects ray class groups and Galois groups.

REPLY [4 votes]: Let $L/K$ be an abelian extension of global fields. If $C_K$ denotes idèle class group of $K$, then global class field theory gives an isomorphism between $C_K/N_{L/K}C_L$ on the one hand, and the Galois group $G_{L/K}$ on the other hand, called Artin's reciprocity law. Using this isomorphism, you can regard any character of $G_{L/K}$ as a character of $C_K$, which gives you the interpretation of Artin $L$-functions as Hecke $L$-functions.
This correspondence is as explicit, as the reciprocity law, i.e. fairly explicit. To define a homomorphism $C_K\rightarrow G_{L/K}$, it is enough to say what it does at each local place, and then extend multiplicatively. Artin's reciprocity map sends the uniformiser $\pi$ at an unramified place $\mathfrak{p}$ to the Frobenius element at $\mathfrak{p}$, i.e. the unique element of $G_{L/K}$ that acts as the Frobenius on the extension of finite fields that you get by reducing $K$ modulo $\mathfrak{p}$ and $L$ modulo a prime above it. There are tricks for computing the Frobenius element at a given prime, but if you want to do it by hand for a few primes in a low-tech way, you just reduce and check what each element of $G_{L/K}$ reduces to.