The Deep Riemann Hypothesis for normalized partial Euler products

Let $K$ be a global field. For all but finitely many primes $\mathfrak p$ of $K$, let $M(\mathfrak p)\in\operatorname{GL}_{r_{\mathfrak p}}(\mathbb C)$ be a unitary matrix, and set $M(\mathfrak p)=0$ at the remaining primes. Write $q_{\mathfrak p}=\#\mathbb F_{\mathfrak p}$, define

$$L(s,M)=\prod_{\mathfrak p}\det\left(1-M(\mathfrak p)q_{\mathfrak p}^{-s}\right)^{-1},$$

and assume that $L(s,M)$ extends to an entire function and satisfies a functional equation relating $s$ and $1-s$. Put $m=\operatorname{ord}_{s=1/2}L(s,M)$ and $\delta(M)=-\operatorname{ord}_{s=1}L(s,M^2)$. The Deep Riemann Hypothesis. The limit

$$\lim_{x\to\infty}\left((\log x)^m\prod_{q_{\mathfrak p}\leq x}\det\left(1-M(\mathfrak p)q_{\mathfrak p}^{-\frac12}\right)^{-1}\right)$$

exists and is non-zero; moreover, it converges to

$$\left.\frac{\sqrt{2}^{\delta(M)}}{e^{m\gamma}m!}L^{(m)}(s,M)\right|_{s=\frac12},$$

where $\gamma$ is the Euler constant and $L^{(m)}(s,M)$ denotes the derivative of order $m$. The paper presents these as DRH (A), concerning existence and non-vanishing, and DRH (B), specifying the value. The hypothesis is used to relate prime-number-race asymptotics to convergence of partial Euler products, but its general validity remains open.