6 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 $\pi$ be a nontrivial cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb A)$. For each prime $v$ with Satake parameter $M_\pi(v)$…

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…

Let $M(p)$ be a unitary matrix of degree $r\in\mathbb N$ for each prime $p$, and let … be an $L$-function that extends to an entire function, satisfies a functional equation with…

Deep Riemann hypothesis. The limit