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 $$\exp \left(\delta \sqrt{\frac{\log D\log \log \log D}{\log \log D}}\right)$$ for any $\delta <\frac{1}{4}$.