Central Values of $L$-Functions of Twisted Modular Forms and Local Polynomials

In this paper we study the product of two central values of $L$-functions of a twisted modular. We show that it suffices to compute a local polynomial at a finite number of points to decide whether the product is zero. For the proof, we relate the local polynomial to the product of the $L$-functions using a locally harmonic Maass form and building on the Shimura-Shintani correspondence. This extends results from Ehlen, Guerzhoy, Kane and Rolen as well as Males, Mono, Rolen and Wagner.