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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.

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    $\begingroup$ The rule of thumb is this: the $L$-function of an irreducible $n$-dimensional Galois representation equals (conjecturally) the $L$-function of an (algebraic) cuspidal representation of $\mathrm{GL}_n$ over $\mathbb{Q}$. That is, in the case of $K$ Galois, only Dirichlet $L$-functions and modular $L$-functions appear in the factorization of $\zeta_K(s)$ if and only if all irreducible representations of $\mathrm{Gal}(K/\mathbb{Q})$ are at most two-dimensional. Note that modular $L$-functions include the $L$-functions of Maass forms. $\endgroup$ – GH from MO Jul 31 '15 at 0:09
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    $\begingroup$ @KevinVentullo: Actually some researchers (like myself) also include Maass forms among modular forms, and distinguish "the older modular forms" by saying "holomorphic modular forms". $\endgroup$ – GH from MO Jul 31 '15 at 0:17
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    $\begingroup$ Okay fair enough, I just wanted to avoid confusion for the OP $\endgroup$ – Kevin Ventullo Jul 31 '15 at 0:18
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    $\begingroup$ @Myshkin: By standard conjectures, the Artin $L$-functions are primitive, so yes, the known instances of factoring a Dedekind zeta are precisely when the Artin $L$-functions occurring in the factorization are known to satisfy Artin's conjecture (i.e. when they are holomorphic). $\endgroup$ – GH from MO Jul 31 '15 at 0:56
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    $\begingroup$ @Myshkin: The "only if" part follows from the first sentence (i.e. the strong Artin conjecture) and the fact that $L$-functions of cuspidal representations are expected to be primitive. For example, if (conjecturally) a $\mathrm{GL}_3$ cuspidal $L$-function is present, you cannot factor it into $\mathrm{GL}_1$ and $\mathrm{GL}_2$ $L$-functions. For your other comment: I don't understand what you mean by a concrete factorization. Artin $L$-functions are pretty concrete in my opinion. $\endgroup$ – GH from MO Jul 31 '15 at 1:47
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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.

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  • $\begingroup$ For non-abelian cubic extension $K=Q[x]/(f), D = disc(f) $( $ < 0$),$ F= Q(\sqrt{D}), L = KF$ then $\zeta_K(s)=\zeta(s)L(s,\rho : Gal(L/Q)\to GL_2)$ where $\rho$ is the dihedral representation of $S_3 = Gal(L/Q)$ and by class field theory or by CM on the elliptic curve with CM by an order of $O_F$ such that $F(j(E))=L$ then $L(s,\rho,Gal(L/Q)\to GL_2)=L(s,\psi : Gal(L/F)\to GL_1)=L(s,\psi,F)$ $\endgroup$ – reuns Feb 3 at 6:11
  • $\begingroup$ where the latter is an Hecke L-function, corresponding to an automorphic representation of $GL_1(F)\setminus GL_2(\mathbb{A}_F)$. In addition $L(s,\psi,F)) = L(s,f)$ with $f(z) = \sum_{a \in O_F } \psi(a) e^{2i \pi |a|^2 z} \in S_{1+2r}(\Gamma_1(disc(O_F) m))$, $m$ the conductor of $\psi$ (the series you wrote, with one quadratic form by ideal class of $\mathbb{Z}+m O_F$) and $|\psi(a)| = |a|^r$ which gives an automorphic representation of $GL_2(\mathbb{Q})\setminus GL_2(\mathbb{A_Q})$. $\endgroup$ – reuns Feb 3 at 8:51

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