The Lerch zeta function is a three variable zeta function, with variables $(s, a, c)$, which generalizes the Riemann zeta function and has a functional equation, but no Euler product. We discuss its properties. It is an eigenfunction of a linear partial differential equation in the variables $(a, c)$ with eigenvalue $-s$, and it is also preserved under a a commuting family of two-variable Hecke-operators $T_m$ with eigenvalue $m^{-s}$. We give a characterization of it in terms of being a simultaneous eigenfunction of these Hecke operators. We then give an automorphic interpretation of the Lerch zeta function in terms of Eisenstein series taking values on the Heisenberg nilmanifold, a quotient of the real Heisenberg group modulo its integer subgroup. Part of this work is joint with W.-C. Winnie Li.