On Landau-Siegel zeros and heights of singular moduli

Let ${\chi }_D$ be the Dirichlet character associated to $\mathbb{Q}(\sqrt{D})$ where $D<0$ is a fundamental discriminant. Improving Granville-Stark [DOI:10.1007/s002229900036], we show that $$\frac{L^{\mathrm{\prime }}}{L}(1,{\chi }_D)=\frac{1}{6}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}(j({\tau }_D))-\frac{1}{2}\log \mathrm{\mid }D\mathrm{\mid }+C+o_{D\to -\mathrm{\infty }}(1),$$ where ${\tau }_D=\frac{1}{2}(-\delta +\sqrt{D})$ for $D \equiv \delta ~($\mathrm{m}\mathrm{o}\mathrm{d}$~4)$and$j(\cdot)$isthe$j$-invariantfunctionwith$C = -1.057770\ldots$\mathrm{.}Assumingthe\mathrm{‘}\mathrm{‘}unifor{m}^{\mathrm{\prime }\mathrm{\prime }}$abc$-conjecture for number fields, we deduce that $L(\beta ,{\chi }_D)\mathrm{\ne }0$ with $\beta \geq 1 - \frac{\sqrt{5}\varphi + o(1)}{\log|D|}$where$\varphi = \frac{1+\sqrt{5}}{2}$, which we improve for smooth $D$.