  The Shimura correspondence connects modular forms of integral weights and
half-integral weights. One of the directions is realized by the Shintani lift,
where the inputs are holomorphic differentials and the outputs are holomorphic
modular forms of half-integral weight. In this article, we generalize this lift
to differentials of the third kind. As an application we obtain a modularity
result concerning the generating series of winding numbers of closed geodesics
on the modular curve.
