In this talk, I would talk about establishing a close connection between three notions attached to a modular subgroup, namely, the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup, and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, I show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.