Convergence conjecture for Artin Euler products at the central point

Let $K$ be a global field and let $\rho$ be an Artin representation with Artin $L$-function $L(s,\rho)$. Set $m=\operatorname{ord}_{s=1/2}L(s,\rho)$. Convergence conjecture. There exists a constant $L$ such that

$$(\log x)^m\prod_{\mathrm N(v)\le x}\det\left(1-\rho(\operatorname{Frob}_v)\mathrm N(v)^{-1/2}\right)^{-1}=L+o(1)$$

as $x\to\infty$. This is a weakened central-point form of the Deep Riemann Hypothesis, asserting convergence after the expected logarithmic normalization; its general validity remains open.