Joint-moment conjecture for derivatives of quadratic Dirichlet L-functions

Let $\mathcal{D}(X)$ be the set of fundamental discriminants $d$ with $|d|<X$, and let $L(s,\chi_d)$ be the Dirichlet $L$-function attached to the quadratic character $\chi_d$. For integers $0\leq n_1\leq n_2$ and non-negative integers $k_1,k_2$ not both zero, set $k=k_1+k_2$. Joint-moment conjecture. As $X\to\infty$,

$$\frac{1}{|\mathcal{D}(X)|}\sum_{d\in\mathcal{D}(X)}L^{(n_1)}(1/2,\chi_d)^{k_1}L^{(n_2)}(1/2,\chi_d)^{k_2}\sim a_k\,b_{k_1,k_2}^{Sp}(n_1,n_2)(\log X)^{k(k+1)/2+k_1n_1+k_2n_2},$$

where

$$a_k=\prod_{p\ \textup{\prime}}\frac{(1-1/p)^{k(k+1)/2}}{1+1/p}\left(\frac{(1-1/\sqrt p)^{-k}+(1+1/\sqrt p)^{-k}}{2}+\frac{1}{p}\right),$$

and $b_{k_1,k_2}^{Sp}(n_1,n_2)$ is the symplectic random-matrix coefficient defined in the paper. This is the symplectic random-matrix prediction for joint moments in the quadratic Dirichlet family; the arithmetic factor $a_k$ is the same one occurring in conjectures for the moments of $L(1/2,\chi_d)$.