elliptic partial differential operator that governs the one-loop vortex fluctuations. This Hessian operator is a matrix second-order partial differential operator (PDO) of Schr$\ddot{\rm o}$dinger type. Its square root is defined in the framework of complex powers of elliptic (pseudo) differential operators, a well developed and sound mathematical theory. The formal trace is then the spectral zeta function of the elliptic PDO exhibiting analytical properties in the complex $s$-plane of the exponent. Nevertheless, use of the zeta function with the purpose of regularizing divergences in QFT requires to dispose of more detailed information about its description. The usual strategy developed by the physicist’s community is to take profit of the more tractable spectral heat function to pass to the zeta function via Mellin transform, see e.g. References [@Elizalde1994; @Vassilevich2003:prc; @Avramidi2002:npp]. In particular it is a common technique in dealing with quantum fields on curved spaces and/or extended/solitonic backgrounds to start from the high-temperature (short time) asymptotic expansion of the heat equation kernel following the seminal works of deWitt [@deWitt1965] and Gilkey [@Gilkey1975:jdg] [[^1]]{}. All this machinery is well behaved if the field fluctuations are strictly $L^2$. In QFT, however, two characteristics of the spectrum of the PDO at the stake disturb 