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Hoffstein and Hulse recently introduced the notion of shifted convolutionDirichlet series for pairs of modular forms $f_1$ and $f_2$. The second twoauthors investigated certain special values of symmetrized sums of suchfunctions, numbers which are generally expected to be mysterious transcendentalnumbers. They proved that the generating functions of these values in$h$-aspect are linear combinations of mixed mock modular forms and quasimodularforms. Here we examine the special cases when $f_1=f_2$ where, in addition,there is a prime $p$ for which $p^2$ divides the level. We prove that the mixedmock modular form is a linear combination of at most two weight 2 weaklyholomorphic $p$-adic modular forms.

$p$-adic properties of modular shifted convolution Dirichlet series