Question

The Dedekind zeta function of an abelian extension $E$ of $\mathbb{Q}$ factors as a product of Artin L function $L(s, \chi)$, where the product runs over all irreducible representations $\chi$ of $Gal(E : \mathbb{Q})$ .

Question: What is known for irreducible representation $ \sigma$ of $G(F) = Gal(\overline{Q}, F)$. How does the Artin $L$ function decompose? Something like $$L_F(s, \sigma) = \prod\limits_{\sigma' \subset Ind_{G(F)}^{G(E)} \sigma} L_E(s, \sigma'),$$ where $F$ is a finite extension of $E$?

Answer 1

You should define what you mean by a decomposition of an Artin $L$-function. If you assume standard conjectures of Langlands and Selberg, then the Artin $L$-function of an irreducible representation of $G(\mathbb{Q})$ is a primitive function in the Selberg class, hence it has no nontrivial decomposition there (or among Artin $L$-functions for that matter). If you start with an irreducible representation $\sigma$ of $G(F)$, then $L_F(s,\sigma)=\prod_\rho L_\mathbb{Q}(s,\rho)^{m(\rho)}$, where $\rho$ runs through the irreducible representations of $G(\mathbb{Q})$ and $m(\rho)$ denotes the multiplicity of $\rho$ in the induced representation of $\sigma$ from $G(F)$ to $G(\mathbb{Q})$. This should be the unique maximal factorization into $L$-functions over $\mathbb{Q}$. In particular, if $F/\mathbb{Q}$ is Galois, then $L_F(s,\sigma)$ should be "irreducible". For a reference I recommend Murty's paper here.

Answer 2

Thinking from the automorphic point of view, it should be an $L$-function over $\mathbb Q$ corresponding to the automorphic induction of the Hecke character to a representation $\Pi$ of $GL_N(\mathbb A_{\mathbb Q})$ where $N=deg(E:\mathbb Q)$. However, automorphic induction is only known for cyclic (whence solvable) and non-normal cubic extensions.

Depending on your extension and what you induce, it may be that $\Pi$ is cuspidal, but in general it should break up as an isobaric sum $$ \Pi = \pi_1 \boxplus \cdots \boxplus \pi_r,$$ where the $\pi_i$'s are cuspidal. ($r=1$ and $\pi_1 = \Pi$ if $\Pi$ is cuspidal). Then your desired $L$-function decomposition over $\mathbb Q$ is $$ L(s,\pi_1)L(s,\pi_2) \cdots L(s,\pi_r).$$

In the case of the Dedekind zeta function for an abelian extension, these $\pi_i$'s correspond to the irreducible characters of Gal$(E/\mathbb Q)$. For more general extensions, one knows this when each irreducible representation of Gal$(E/\mathbb Q)$ is known to be modular (corresponding to a cuspidal automorphic representation).