A convergent sum involving the zeta function

Question

Let $\rho = \beta+i\gamma$ be a non-trivial zero of $\zeta(s)$. Davenport (see bottom of page 81) claims that the sum $$\sum_{\rho} \frac{1}{\rho}$$ converges because one can group together the terms between $\rho$ and its complex conjugate $\bar{\rho}$ such that $$\sum_{\rho} \frac{1}{\rho}=\sum_{\gamma>0} \left( \frac{1}{\rho}+ \frac{1}{\bar{\rho}} \right) \leq \sum_{\gamma>0} \frac{2}{\vert \rho \vert^2},$$ which is known to converge. Why can the sum $\sum_{\rho} \frac{1}{\rho}$ be added term by term such that the terms involving $\rho$ and $\bar{\rho}$ can be paired together? Wouldn't we already have to assume that the sum converges unconditionally to show that its terms can be rearranged?