Artin Holomorphy Conjecture

Let \(L/K\) be a finite Galois extension of number fields with group \(G\), and let \(\rho:G\to\mathrm{GL}(V)\) be a nontrivial irreducible finite-dimensional complex representation. For every nonzero prime ideal \(\mathfrak p\) of \(K\), choose a prime of \(L\) above it, let \(I_{\mathfrak p}\) be its inertia group, and let \(\operatorname{Frob}_{\mathfrak p}\) denote an arithmetic Frobenius element acting on \(V^{I_{\mathfrak p}}\). The Artin \(L\)-function
\[
  L(s,\rho,L/K)=
  \prod_{\mathfrak p}
  \det\!\left(1-\rho(\operatorname{Frob}_{\mathfrak p})
       N\mathfrak p^{-s}\mid V^{I_{\mathfrak p}}\right)^{-1},
  \qquad \Re s>1,
\]
has an analytic continuation to an entire function on \(\mathbb C\).