I was wondering whether there is (a known) upper bound of the order of the non-trivial zeros of Dirichlet $L$-functions. For a zero $s$ of the Riemann zeta function we have the estimate $C\cdot\log(\operatorname{Im}(s))$ where $C$ is some positive constant (Do we know an upper bound for the multiplicity of the non-trivial zeros of Zeta?).
For the non-trivial zeros the order should be $1$, based on the fact that the gamma function has simple poles at the negative integers (derived from the functional equation for those $L$ functions).
An estimate similar to that of the zeta function would be rather helpful to show the convergence of a series ranging over the reciprocals of the (non-trivial) zeros squared! (counting multiplicity :P)