Starting from Montgomery’s conjecture, there has been a substantial interest on the connections of random matrix theory and the theory of L-functions. In particular, moments of characteristic polynomials of random matrices have been considered in various works to estimate the asymptotics of moments of L-function families. In this paper, we first consider joint moments of the characteristic polynomial of a symplectic random matrix and its second derivative. We obtain the asymptotics, along with a representation of the leading order coefficient in terms of the solution of a Painlevé equation. This gives us the conjectural asymptotics of the corresponding joint moments over families of Dirichlet L-functions. In doing so, we compute the asymptotics of a certain additive Jacobi statistic, which could be of independent interest in random matrix theory. Finally, we consider a slightly different type of joint moment that is the analogue of an average considered over $U(N)$ in various works before. We obtain the asymptotics and the leading order coefficient explicitly.

从蒙哥马利猜想开始，人们对随机矩阵理论和L函数理论的联系产生了浓厚的兴趣。特别是，随机矩阵的特征多项式的矩已在各种工作中被考虑来估计 L 函数族矩的渐近性。在本文中，我们首先考虑辛随机矩阵的特征多项式及其二阶导数的联合矩。我们获得了渐近方程，以及以 Painlevé 方程的解表示的首阶系数。这给了我们狄利克雷 L 函数族上相应联合矩的猜想渐近性。在此过程中，我们计算了某个加性雅可比统计量的渐近，这在随机矩阵理论中可能具有独立的意义。最后，我们考虑一种稍微不同类型的关节力矩，它类似于之前各种工作中考虑的 $U(N)$ 平均值。我们明确地获得了渐进系数和首阶系数。