Local factor of an Artin L-function ($L_{\mathfrak{p}}(\rho ,s)=\det (1-(N\mathfrak{p}{)}^{-s}\rho ({\mathrm{Frob}}_{\mathfrak{p}}){|}_{V^{I_{\mathfrak{p}}}}{)}^{-1}$)

Only the inertia group invariants enter an Artin L-function local factor. A Frobenius automorphism lift acts unambiguously on this subspace, since other lifts differ by inertia. For an unramified one-dimensional group representation, the factor is $(1-\chi ({\mathrm{Frob}}_{\mathfrak{p}})(N\mathfrak{p}{)}^{-s}{)}^{-1}$; for a ramified one-dimensional group representation, the invariant subspace is zero and the factor is one. This explains the zero values used at conductor primes by quadratic Dirichlet characters.