Shifted second moment of Gaussian Hecke $L$-functions $L(s,\lambda^k)$

Abstract

We establish a uniform asymptotic formula for the smoothly weighted shifted second moment of the Gaussian angular Hecke $L$-functions $L(s,\lambda^k)$. The completed moment is expressed as the sum of four explicit main terms, corresponding to the four functional-equation swaps, with an error of size $O_{\Phi,\epsilon}(K^{1/2+\epsilon})$.After the Archimedean factors are removed, the resulting formula agrees with the numerator-only specialization of the four-swap prediction of the $L$-functions Ratios Conjecture. The proof transforms the off-diagonal contribution into Weyl sums over the roots of $r^2\equiv-1\pmod C$, realizes these sums spectrally through incomplete Poincar\'e series evaluated at $i$, and separates the Eisenstein and Maa\ss{} spectra. The two $v$-type main terms arise respectively from the zero frequency and from the combined residues of two moving Eisenstein poles, while the cuspidal spectrum is absorbed into the square-root error term.