This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by $\exp(C_K |s|^{\delta})$ in strips $|\Real s| \leq K$, where $\delta$ is the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott--Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions $\{|\Real s | \leq |\Imag s|^\alpha\}$ is given, followed by weaker lower bound estimates in strips $\{\Real s > -C, |\Imag s|\leq r\}$, and logarithmic neighbourhoods $\{ |\Real s | \leq \rho \log |\Imag s| \}$. Recent numerical work of Strain--Zworski suggests the upper bounds in strips are optimal.