Pasol–Stevens conjecture on the evil Eisenstein eigensymbol

Let $p$ and $\ell$ be primes, let

$$E_{2,\ell}=\frac{\ell-1}{24}+\sum_{n\geq 1}a_nq^n\in M_2(\Gamma_0(\ell)),$$

where

$$a_n=\sum_{\substack{d\mid n\\\\ell\nmid d}}d.$$

Let $\Phi\in\operatorname{Symb}^+_{\Gamma_0(p\ell)}({\mathcal D}_0)$ be an overconvergent eigensymbol with the same Hecke eigenvalues as the critical slope refinement of $E_{2,\ell}$, and write $L_p(\Phi,s)$ for its $p$-adic $L$-function and $\operatorname{wt}(s)$ for the weight factor. Pasol–Stevens conjecture. The $E_{2,\ell}$-eigenspace of $\operatorname{Symb}^+_{\Gamma_0(p\ell)}({\mathcal D}_0)$ is $1$-dimensional, generated by $\Phi$, and

$$L_p(\Phi,s)=\operatorname{wt}(s)(1-\ell^s)\zeta_p(s+1)\zeta_p(1-s).$$

This conjecture arose from numerical experiments for $p=3$ and $\ell=11$ and predicts both the uniqueness of the relevant overconvergent eigensymbol and an explicit factorization of its $p$-adic $L$-function.