$p$-adic moments of $L$-functions

We obtain a formula for the $p$-adic valuation of weighted moments of central $L$-values of holomorphic cusp forms twisted by Dirichlet characters of order $p$. In some cases we give an arithmetic interpretation of the constants in the formula. The result is obtained via the study of the digit map, turning a horizontal $p$-adic measure into a vertical one, applied to the horizontal $p$-adic $L$-functions as defined by the authors in previous work.