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.