It's simply impossible to study modular forms without any complex analysis. Generally, a lot of analytic number theory depends on complex analysis.

Elliptic curves can be studied either via algebraic geometry or (over C) via complex analysis.

So my recommendation would be a book that's not actually a book on analytic number theory per se: Freitag's Complex Analysis. This discusses modular forms, elliptic curves (via elliptic functions) and some other analytic NT such as the functional equation for the Riemann zeta function, while also giving you the necessary background from complex analysis.