= Local factor of an Artin L-function
{title2=$L_{\mathfrak p}(\rho,s)=\det(1-(N\mathfrak p)^{-s}\rho(\operatorname{Frob}_{\mathfrak p})|_{V^{I_{\mathfrak p}}})^{-1}$}
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 $(1-\chi(\operatorname{Frob}_{\mathfrak p})(N\mathfrak p)^{-s})^{-1}$; 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>.
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