The usual formulation of Artin reciprocity (for example in Neukirch or Lang's book on algebraic number theory), applied to L-funcions, says that for an irreducible character $\chi$ of an abelian extension, $$L(\chi,s)=L(\tilde{\chi},s)$$ where $\tilde{\chi}$ is a Grobencharakter, and the L-series on the right is an abelian (Hecke) L-funcion.
But another formulation is that Artin reciprocity implies the equality above for all 1-dimensional characters.
How does one prove this?
I'm sure this is an easy consequence, but I don't understand it. It is easy to see that:
abelian extension $\to$ 1-dimensional character
But the converse is not true, there are plenty of 1-dimensional characters of non-abelian extensions.