The Artin $L$-series for Abelian extensions are known to coincide with Hecke $L$-series, which in particular implies that if $E/K$ is a Abelian extensions, and $\chi$ is a non-trivial simple character of $\textrm{Gal}(E/K)$, then $L(E/K,\chi,s)$ admits an analytic continuation holomorphic on $\mathbb{C}$. This proves the Artin conjecture for Abelian extensions.
It is said that this settles the Artin conjecture for all degree 1 representations. But how?
Say we have $\textrm{Gal}(E/K) \cong S_3$. How does the above fact imply that $L(E/K,\chi,s)$ is entire for any non-trivial simple degree 1 character $\chi$?
All help or input would be highly appreciated.
Best Answer
Let $G=\textrm{Gal}(E/K)$
By the universal property of the abelianization, $\chi:G \to \Bbb C^\times$ factors over $G^{ab}$, so we obtain $\overline{\chi}:\textrm{Gal}(L/K) \to \Bbb C^\times$ where $L=E^{[G,G]}$. Now $L/K$ is abelian and $\chi$ is the inflation of $\overline{\chi}$, so by functoriality of Artin L-functions with respect to inflations we have $$L(E/K,\chi,s)=L(L/K,\overline{\chi},s)$$