Gross's first-part conjecture on orders of vanishing

Let $\psi\in\operatorname{Irr}_{\mathbb C_p}(\mathcal G)$ and let $r_S(\psi)$ be the order of vanishing at $s=0$ of the corresponding complex $S$-truncated Artin $L$-function. Gross's first-part conjecture. The order of vanishing at $s=0$ of $L_{p,S}(s,\psi)$ equals $r_S(\psi)$. This conjecture compares the analytic behavior of $p$-adic and complex Artin $L$-functions. It is established for some classes of characters, but remains open in general.