Relationship between different L-functions

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 :

1. Artin's Reciprocity Laws.
2. A fantastic description of Artin's-L-Functions by J.W.Cogdell which is here.
3. 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.

Let $L/K$ be an abelian extension of global fields. If $C_K$ denotes idèle class group of $K$, then global class field theory gives an isomorphism between $C_K/N_{L/K}C_L$ on the one hand, and the Galois group $G_{L/K}$ on the other hand, called Artin's reciprocity law. Using this isomorphism, you can regard any character of $G_{L/K}$ as a character of $C_K$, which gives you the interpretation of Artin $L$-functions as Hecke $L$-functions.

This correspondence is as explicit, as the reciprocity law, i.e. fairly explicit. To define a homomorphism $C_K\rightarrow G_{L/K}$, it is enough to say what it does at each local place, and then extend multiplicatively. Artin's reciprocity map sends the uniformiser $\pi$ at an unramified place $\mathfrak{p}$ to the Frobenius element at $\mathfrak{p}$, i.e. the unique element of $G_{L/K}$ that acts as the Frobenius on the extension of finite fields that you get by reducing $K$ modulo $\mathfrak{p}$ and $L$ modulo a prime above it. There are tricks for computing the Frobenius element at a given prime, but if you want to do it by hand for a few primes in a low-tech way, you just reduce and check what each element of $G_{L/K}$ reduces to.