  Background $\alpha$-nucleus potentials play an essential role for the
calculation of $\alpha$-induced reaction cross sections at low energies in the
statistical model. Uncertainties of these calculations are related to
ambiguities in the adjustment of the potential parameters to experimental
elastic scattering angular distributions (typically at higher energies) and to
the energy dependence of the effective $\alpha$-nucleus potentials.
  Purpose The present work studies cross sections of $\alpha$-induced reactions
for $^{64}$Zn at low energies and their dependence on the chosen input
parameters of the statistical model calculations. The new experimental data
from the recent Atomki experiments allow for a $\chi^2$-based estimate of the
uncertainties of calculated cross sections at very low energies.
  Method The recent data for the ($\alpha$,$\gamma$), ($\alpha$,$n$), and
($\alpha$,$p$) reactions on $^{64}$Zn are compared to calculations in the
statistical model. A survey of the parameter space of the widely used computer
code TALYS is given, and the properties of the obtained $\chi^2$ landscape are
discussed.
  Results The best fit to the experimental data at low energies shows $\chi^2/F
\approx 7.7$ per data point which corresponds to an average deviation of about
30% between the best fit and the experimental data. Several combinations of the
various ingredients of the statistical model are able to reach a reasonably
small $\chi^2/F$, not exceeding the best-fit result by more than a factor of 2.
  Conclusions The present experimental data for $^{64}$Zn in combination with
the statistical model calculations allow to constrain the astrophysical
reaction rate within about a factor of 2. However, the significant excess of
$\chi^2/F$ of the best-fit from unity asks for further improvement of the
statistical model calculations and in particular the $\alpha$-nucleus
potential.
