  In 2017, Ono, Rolen, and Sprung [ORS17] answered problems of Manin [Man16] by
defining zeta-polynomials $Z_f(s)$ for even weight newforms $f\in
S_k(\Gamma_0(N)$; these polynomials can be defined by applying the
"Rodriguez-Villegas transform" to the period polynomial of $f$. It is known
that these zeta-polynomials satisfy a functional equation $Z_f(s) = \pm
Z_f(1-s)$ and they have a conjectural arithmetic-geometric interpretation.
Here, we give analogous results for a slightly larger class of polynomials
which are also defined using the Rodriguez-Villegas transform.
