Extreme values of class group $L$-functions

We give a lower bound for the maximum value of class group $L$-functions attached to $\mathbb{Q}(\sqrt{-D})$ at the central point and show that this value is on average at least $$\exp\Bigg(\delta\sqrt{\frac{\log D \log \log \log D}{\log \log D}} \Bigg)$$ for any $\delta < \frac{1}{4}$.