Your question is a very good one, To add something I know it appears as follows :
" Its a well known result that Artin , through his famous " Reciprocity law " has proved that in the case of abelian extensions of number fields the Artin L-functions are Hecke-L-functions.
To say further Let $E⁄K$ be an abelian Galois extension with Galois group $G$. Then for any character \sigma: G \to C^× (i.e. one-dimensional complex representation of the group $G$), there exists a Hecke character $\chi$ of $K$ such that
$L^{\large\rm{Artin}}_{E/K}(\sigma,s)=L^{\large\rm{Hecke}}_{K}(\chi,s)$
where the left hand side is the Artin $L$-function associated to the extension with character $\sigma$ and the right hand side is the Hecke $L$-function associated with $\chi$, The formulation of the Artin reciprocity law as an equality of $L$-functions allows formulation of a generalisation to n-dimensional representation, though a direct correspondence still lacking.
And to mention the whole proof is completely difficult and strenuous too, I try to mention the gist and big-picture.
A Hecke L-function as everyone know is an Euler product $L(s, χ)$ attached to a character $χ$ of F^× \backslash I_F where $I_F$ is the group of ideles of $F$ . If $v$ is a place of $F$ then F^{×}_{v} imbeds in $I_F$ and $χ$ defines a character $χ_v$ of F^{×}_{v} . To form the function $L(s, χ)$ we take a product over all places of $F$:
$L(s, χ) = \prod _v L(s, χ_v).$
An Artin $L$-function is associated to a finite-dimensional representation $\rho$ of a Galois group $\rm{Gal}(K/F)$, $K$ being an extension of finite degree. It is defined arithmetically and its analytic properties are extremely difficult to establish. Once again $L(s,\rho ) = L(s, \rho_v),$ $\rho_{v}$ being the restriction of to the decomposition group. For our purposes it is enough to define the local factor when $v$ is defined by a prime $p$ and $p$ is unramified in $K$. Then the Frobenius conjugacy class $Φ_p$ in $\rm{Gal}(K/F)$ is defined, and $L(s, \rho_{v}) = \frac{1}{\large det(I − \rho(Φ_p)/Np^s)} = \prod^{d}_{i=1}\frac{1}{\large 1 − β_i(p)/Np^s} $, if $β_1(p), . . . , β_d(p)$ are the eigenvalues of $\rho(Φ_p)$.
Although the function $L(s,\rho )$ attached to $\rho$ is known to be meromorphic in the whole plane, Artin’s conjecture that it is entire when $\rho$ is irreducible and nontrivial is still outstanding. Artin himself showed this for one dimensional , and it can now be proved that the conjecture is valid for tetrahedral $\rho$ , as well as a few octahedral $\rho$ ( see the references I have searched in google and provided in below section ) . Artin’s method is to show that in spite of the differences in the definitions the function $L(s,\rho )$ attached to a one-dimensional $\rho$ is equal to a Hecke $L$-function $L(s, χ)$ where $χ = χ(\rho)$ is a character of F^× \backslash I_F . He employed all the available resources of class field theory, and went beyond them, for the equality of $L(s,\rho )$ and $L(s, χ(\rho))$ for all $\rho$ is pretty much tantamount to the Artin reciprocity law, which asserts the existence of a homomorphism for $I_F$ onto the Galois group $\rm{Gal}(K/F)$ of an abelian extension which is trivial on F^× and takes $\omega_{p}$ to $Φ_p$ for almost all $p$.
The equality of $L(s,\rho )$ and $L(s, χ)$ implies that of $χ(\omega_p)$ and $\rho(Φ_p)$ for almost all $p$.
So you need to go through these things first :
- Artin's Reciprocity Laws.
- A fantastic description of Artin's-L-Functions by J.W.Cogdell which is here.
- Interesting proofs found at papers of Artin "Uber eine neue Art von L-Reihen " and " Zur theorie der L-Reihen mit allgemeinen Gruppencharackteren".
Thats all I know,
Thank you.