It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this:
$$\zeta_K(s)=\zeta(s)\prod_{\chi \neq 1} L(s,\chi)$$
with the Dirichlet characters distinct and primitive.
If $K$ is non-abelian but Galois, we instead have (by Aramata-Brauer):
$$\zeta_K(s)=\zeta(s)\prod_{\rho \neq 1} L(s,\rho)$$
with the representations non-trivial and irreducible, although in this case we don't know unconditionally that that's a factorization of irreducible L-functions. In the non-Galois case, factors might appear with arbitrary integer powers. But let's forget about that for a moment.
In this answer, Kevin Dong mentions an explicit factorization of the zeta of $\mathbb{Z}[\sqrt[3]{2}]$ in terms of modular forms. It is very nice but not quite suprising: Artin L-functions are expected (and known in some cases) to always be automorphic.
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I'm interested on the proof for $\mathbb{Z}[\sqrt[3]{2}]$ (I haven't been able to find a reference for it), and any other reference for known case of such a factorization (this is, not a factorization on terms of Artin L-function, but of automorphic ones).
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I'd also want to know what the conjectures are (on the automorphic side) for what the factorization looks like for an arbitrary non-Galois Dedekind zeta function.
Any other information around those issues might be of help, but not generally about Langlands or the Artin conjecture.
Best Answer
if you know theta series, you write $L(s,\rho)= \sum_{n>0 } a_n n^{-s}$ and you have ; $\sum_{n>0} a_nq^n = 1/2(\Theta_{1,0,27}-\Theta_{4,2,7})$. Where : $$ \Theta_{a,b,c} = \sum_{(x,y) \in \mathbb{Z}^2} q^{ax^2+bxy+cy^2} $$
For diedral odd representation all is explicit.