The $p$-adic Stark conjecture at $s=1$

Let $E/F$ be a finite Galois extension of totally real number fields, with $G=\operatorname{Gal}(E/F)$. Let $p$ be a prime, let $\Sigma$ be a finite set of places of $F$ containing $\Sigma_{p}(F)\cup\Sigma_{\infty}(F)$, and let $\rho\in R_{\mathbb{C}_{p}}^{+}(G)$. For a field isomorphism $j:\mathbb{C}\cong\mathbb{C}_{p}$, let $\Omega_{j}(\rho)$ denote the period factor in the conjecture. The $p$-adic Stark conjecture at $s=1$. For every choice of $j$,

$$L_{p,\Sigma}^{*}(1,\rho)=\Omega_{j}(\rho)\,j\left(L_{\Sigma}^{*}(1,\rho^{j^{-1}})\right).$$

This conjecture compares the leading term of the $p$-adic Artin $L$-function with the complex leading term through a period. The source attributes earlier formulations to Serre and Tate and presents this variant for applications; no resolution is supplied.