Negative discrete second moments of Dirichlet $L$-functions

Let $\chi$ be a primitive Dirichlet character modulo $q>1$. Assuming the Generalised Riemann Hypothesis for $L(s,\chi )$ and that the non-trivial zeros $\rho =\frac{1}{2}+i\gamma$ of $L(s,\chi )$ are simple, we prove lower bounds for the discrete moments ${\sum }_{0<\gamma \le T}\mathrm{\mid }{L}^{\mathrm{\prime }}(\rho ,\chi ){\mathrm{\mid }}^{-2}$ and ${\sum }_{0<\gamma \le T}\mathrm{\mid }L(2\rho ,{\chi }^2)\mathrm{/}{L}^{\mathrm{\prime }}(\rho ,\chi ){\mathrm{\mid }}^2$, uniformly in the conductor. The bounds capture the proportion $\beta \mathrm{/}(1+\beta )$ of the conjectured asymptotics, where $\beta =\log T\mathrm{/}\log qT$: this is one half whenever $\log q=o(\log T)$, recovering for fixed $q$ the Dirichlet analogues of theorems of Milinovich and Ng and of Sinha, and degrades to $1\mathrm{/}(2+A)$ when $q=T^A$. We conjecture the true leading order asymptotics and their analogues when we average over the family of primitive characters modulo $q$.