An asymptotic expansion for a Lambert series associated to Siegel cusp forms of degree n

Utilizing inverse Mellin transform of the symmetric square L-function attached to Ramanujan tau function, Hafner and Stopple proved a conjecture of Zagier, which states that the constant term of the automorphic function y^12| (z) |² i. e. , the Lambert series y^12₍=₁^ (n) ² e^-4 n y can be expressed in terms of the non-trivial zeros of the Riemann zeta function. This study examines certain Lambert series associated to Siegel cusp forms of degree n twisted by a character and observes a similar phenomenon.