Conjecture on joint moments of quadratic Dirichlet L-functions and their second derivatives

Let $s+\frac{1}{2}>h\geq 0$. For each quadratic Dirichlet character $\chi_d$ with modulus $d\leq N$, let $L(s,\chi_d)$ be its Dirichlet $L$-function, and let $N^*$ denote the number of such characters. Define

$$a_s=\prod_{p\text{ prime}}\frac{\left(1-\frac{1}{p}\right)^{s(s+1)/2}}{1+\frac{1}{p}}\left(\frac{1}{p}+\frac{1}{2}\left(\left(1+\frac{1}{\sqrt p}\right)^{-s}+\left(1-\frac{1}{\sqrt p}\right)^{-s}\right)\right).$$

Joint-moment conjecture. As $N\to\infty$,

$$\frac{1}{N^*}\sum\left|L\left(\frac{1}{2},\chi_d\right)\right|^{2s-h}\left|L”\left(\frac{1}{2},\chi_d\right)\right|^h\sim \log^{s(s+1)/2+2h}(N^{1/2})a_s\frac{2^{s^2/2}G(1+s)\sqrt{\Gamma(1+s)}}{\sqrt{G(1+2s)}\Gamma(1+2s)}\mathbb{E}\left[\left(\frac{1}{2}M\left(s+\frac{1}{2},\frac{1}{2}\right)+1\right)^h\right].$$

The conjecture concerns joint moments of quadratic Dirichlet $L$-functions and their second derivatives, connecting number-theoretic moments with random-matrix predictions involving the random variable $M$.