  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 Painleve 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 the unitary group in
various works before. We obtain the asymptotics and the leading order
coefficient explicitly.
