Let $K$ be a number field, $\chi : C_K \to \mathbb{C}^\ast$ a Hecke character (that is, a character of the idèle class group), and $L(\chi,s)$ the corresponding Hecke $L$-series. I wish to understand how one may construct a Galois extension $G = Gal(L/K)$ and a complex representation $\rho : G \to \mathbb{C}^\ast$ such that $\mathcal{L}(L/K,\rho,s) = L(\chi,s)$, where $\mathcal{L}$ here denotes the Artin L-function.

I know how this works when $\chi$ factors through a congruence subgroup mod $\mathfrak{m}$, or equivalently, if $\chi$ is a Dirichlet character mod $\mathfrak{m}$; namely, take $L = K^\mathfrak{m}$, the ray class field mod $\mathfrak{m}$, and use the Artin symbol to turn the given character into a Galois character.

But I am worried that trying to do the same thing for general $\chi$, replacing the Artin symbol with the map $\phi_K : C_K \to Gal(K^{ab}/K)$, will not work. Indeed, the Wikipedia article on 'Hecke Character' suggests that only the Dirichlet characters are accounted for by Class Field Theory (see the last paragraph in the section 'Definition using ideals'). This worries me.

finite orderHecke character. I had seen alternatively Dirichlet characters to be multiplicative functions on field elements (not ideals), so was confused $\endgroup$