The non-primitive Selberg-class moment conjecture

Let $L(s)=\prod_{j=1}^mL_j(s)^{e_j}$, where the $L_j(s)$ are distinct primitive members of the Selberg class, $e_j\in\mathbb{N}$, and let $Q_j$ and $d_j$ be the conductor parameter and degree in the functional equation of $L_j$. Let $\alpha_{L,k}(n)$ be the Dirichlet coefficients of $L(s)^k$, and set $n_L=\sum_{j=1}^m e_j^2$. Define

$$a_L(k)=\prod_p\left(1-\frac{1}{p}\right)^{n_Lk^2}\sum_{n=0}^{\infty}\frac{|\alpha_{L,k}(p^n)|^2}{p^n}.$$

The non-primitive Selberg-class moment conjecture. For $k>-1/2$,

$$\frac{1}{T}\int_0^T\left|L\left(\frac{1}{2}+it\right)\right|^{2k}dt\sim a_L(k)\prod_{j=1}^m\frac{G^2(e_jk+1)}{G(2e_jk+1)}\left(\log\big(Q_jT^{d_j}\big)\right)^{(e_jk)^2}.$$

This extends the standard moment recipe from primitive to products of distinct primitive Selberg-class $L$-functions, retaining separate random-matrix factors and a combined arithmetic factor. It is presented as a conjectural generalization and is not proved in the paper.