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.
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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. 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. |
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