Special Values of Shifted Dirichlet Series from an Adjoint Map on Almost Holomorphic Modular Forms

We find the adjoint of a map on the space of almost holomorphic modular forms, analogous to the generalized Ramanujan-Serre derivative on the space of quasi-modular forms. As applications, we explicitly find special values of certain shifted Dirichlet series associated to quasi-modular forms and prove a rationality result for the shifted Dirichlet series.