4 problems

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…

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…

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…

Deep Riemann hypothesis. The limit