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.

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).

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.