The relation between the Alexander polynomial and the Lefschetz (spectral) zeta function is analogous to that between the Iwasawa polynomial and the p-adic analytic zeta function. See (Morishita 12, chapter 12).

A generator of the characteristic ideal of (the Pontryagin dual of the Selmer’s subgroup) $Sel(X_\inf ,\rho)^\ast$, called the twisted Iwasawa polynomial, is an analogue of the twisted Alexander polynomial.