Partition values and central critical values of certain modular $L$-functions

Let $p(n)$ denote the number of partitions of a positive number $n$, let $\ell \in \{5,7,11\}$ and let $\delta_\ell$ be the least non-negative residue of $24^{-1}$ modulo $\ell$. In this paper we prove congruences modulo $\ell$ between $\frac{p(\ell n+ \delta_\ell)}{\ell}$ and ratios of central critical values of $L$-functions associated to twists of certain integer weight newforms. In 1999, Guo and Ono proved analogous results for $13\leq \ell \leq 31$.