The L-function associated to an Artin representation $\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}(V)$ is given by an Euler product of the form $\prod_p\det(I-\rho(\mathrm{Frob}(p))p^{-s})^{-1}$, where $\mathrm{Frob}(p)$ denotes the conjugacy class of Frobenius elements above $p$. When $\rho$ is ramified at $p$, $\rho(\mathrm{Frob}(p))$ should be interpreted as acting on the invariant subspace $V^{I_p}$ of the inertia group at $p$.

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