  We compute the adjoint of the Serre derivative map with respect to the
Petersson scalar product by using existing tools of nearly holomorphic modular
forms. The Fourier coefficients of a cusp form of integer weight $k$,
constructed using this method, involve special values of certain shifted
Dirichlet series associated with a given cusp form $f$ of weight $k+2$. As
application, we get an asymptotic bound for the special values of these shifted
Dirichlet series and also relate these special values with the Fourier
coefficients of $f$. We also give a formula for the Ramanujan tau function in
terms of the special values of the shifted Dirichlet series associated to the
Ramanujan delta function.
