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 $E$?



Decomposition of Artin L functions
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](http://www.ams.org/journals/bull/1994-31-01/S0273-0979-1994-00479-3/home.html).


