Deep Riemann hypothesis for Artin L-functions

Let $K$ be a global field, and let

$$\rho:\operatorname{Gal}(K^{\mathrm{sep}}/K)\to\operatorname{Aut}_{\mathbb C}(V),\qquad \rho\neq\mathbf 1,$$

be an $n$-dimensional nontrivial irreducible Artin representation. Write $L_K(s,\rho)$ for its Artin $L$-function, let $V^{I_{\mathfrak p}}$ denote the inertia-invariant subspace at a prime ideal $\mathfrak p$, and set $m=m_\rho=\operatorname{ord}_{s=1/2}L_K(s,\rho)$. Define

$$\nu(\rho)=\operatorname{mult}(\mathbf 1,\operatorname{sym}^{2}\rho)-\operatorname{mult}(\mathbf 1,\wedge^{2}\rho)\in\mathbb Z,$$

where $\operatorname{mult}(\mathbf 1,\sigma)$ is the multiplicity of the trivial representation in $\sigma$.

Deep Riemann hypothesis. The limit

$$\lim_{x\to\infty}(\log x)^m\prod_{N(\mathfrak p)\leq x}\det\left(1-\rho(\operatorname{Frob}_{\mathfrak p})N(\mathfrak p)^{-1/2}\mid V^{I_{\mathfrak p}}\right)^{-1}$$

exists and is nonzero, and satisfies

$$\lim_{x\to\infty}(\log x)^m\prod_{N(\mathfrak p)\leq x}\det\left(1-\rho(\operatorname{Frob}_{\mathfrak p})N(\mathfrak p)^{-1/2}\mid V^{I_{\mathfrak p}}\right)^{-1} =\frac{\sqrt{2}^{\nu(\rho)}L_K^{(m)}(1/2,\rho)}{e^{m\gamma}m!}.$$

This hypothesis gives a precise central-value asymptotic for Euler products of nontrivial irreducible Artin representations and is used in the paper to derive Chebyshev-bias results. The source assumes it in its main arguments; its general validity is not established.