L-series of Harmonic Maass Forms

Abstract

We introduce Harmonic Maass forms and present some of their analytic properties. We continue to define their related L-series by using Laplace transform, and prove their functional equations. Our primary objective is to develop a converse theorem for these L-series in both integral and non-integral weights. This became achievable through the definition of the mentioned L-series on a broader class of test functions. To illustrate the idea, we first present an outline of the special case of weakly holo- morphic modular forms on SL2(Z) and then extend it to Harmonic Maass forms. Subsequently, we consider an example of using the converse theorem.