The exceptional zero conjecture for $p$-adic $L$-functions of motives

Let $M$ be a $p$-ordinary motive, let $L_p(s,M,\chi)$ be its analytic $p$-adic $L$-function, and let $a_0$ be a critical integer with character $\chi_0$ such that the interpolation factor $\mathcal{E}(a_0,M,\chi_0)=0$ and $L(a_0,M,\chi_0)\neq 0$. Let $\mathcal{E}^+(a_0,M,\chi_0)$ be the interpolation factor with its vanishing factors removed, and let $e$ be the order of vanishing of $\mathcal{E}$. Exceptional zero conjecture, vaguely. There should be an analytic $L$-invariant $\mathcal{L}_p^{\operatorname{an}}(a_0,M,\chi_0)\in\mathbf{C}_p^\times$ such that

$$\lim_{s\rightarrow a_0}\frac{L_p(s,M,\chi_0)}{(s-a_0)^e}=\mathcal{L}_p^{\operatorname{an}}(a_0,M,\chi_0)\mathcal{E}^+(a_0,M,\chi_0)\frac{L(a_0,M,\chi_0^{-1})}{\Omega_{a_0,M,\chi_0}},$$

and an arithmetic $L$-invariant $\mathcal{L}_p^{\operatorname{arith}}(a_0,M,\chi_0)\in\mathbf{C}_p$, defined in terms of the arithmetic of $M$, such that

$$\mathcal{L}_p^{\operatorname{an}}(a_0,M,\chi_0)=\mathcal{L}_p^{\operatorname{arith}}(a_0,M,\chi_0).$$

This conjecture predicts both the leading term at a trivial zero of a $p$-adic $L$-function and its interpretation through arithmetic invariants. In the situation studied in the paper, the authors prove the corresponding exceptional zero formula and the equality of the analytic and arithmetic $L$-invariants.