Kurokawa's Deep Riemann Hypothesis for Artin L-functions

Let $K$ be a global field and let $\rho\colon\operatorname{Gal}(K^{\mathrm{sep}}/K)\to\operatorname{GL}_n(\mathbb C)$ be a nontrivial irreducible representation. For a prime $v$ of $K$, write $\operatorname{Frob}_v$ for its Frobenius conjugacy class and $\mathrm N(v)$ for its norm. Kurokawa's Deep Riemann Hypothesis. The partial Euler product converges to the stated values: for $\operatorname{Re}(s)>1/2$,

$$\lim_{x\to\infty}\prod_{\mathrm N(v)\le x}\det(1-\rho(\operatorname{Frob}_v)\mathrm N(v)^{-s})^{-1}=L(s,\rho)\ne0,$$

and, if $\operatorname{Re}(s)=1/2$ and $m$ is the order of vanishing of $L(s,\rho)$ at $s$, then

$$\lim_{x\to\infty}(\log x)^m\prod_{\mathrm N(v)\le x}\det(1-\rho(\operatorname{Frob}_v)\mathrm N(v)^{-s})^{-1}=\frac{L^{(m)}(s,\rho)}{e^{m\gamma}m!}\begin{cases}\sqrt{2}^{\nu},&s=1/2,\\1,&\text{otherwise},\end{cases}$$

where $\nu(\rho)=m(\operatorname{sym}^2\rho)-m(\wedge^2\rho)\in\mathbb Z$ and $m(\rho)$ is the multiplicity of $\mathbf1$ in $\rho$. This is a deep strengthening of the Riemann hypothesis, prescribing convergence of partial Euler products on and to the right of the critical line; it remains open in general.