  We study Frobenius eigenvalues of the compactly supported rigid cohomology of
a variety defined over a finite field of $q$ elements via Dwork's method. A
couple of arithmetic consequences will be drawn from this study. As the first
application, we show that the zeta functions for finitely many related affine
varieties are capable of witnessing all Frobenius eigenvalues of the rigid
cohomology of the variety up to Tate twist. This result does not seem to be
known for $\ell$-adic cohomology. As the second application, we prove several
$q$-divisibility lower bounds for Frobenius eigenvalues of the rigid cohomology
of the variety in terms of the multi-degrees of the defining equations. These
divisibility bounds for rigid cohomology are generally better than what is
suggested from the best known divisibility bounds in $\ell$-adic cohomology,
both before and after the middle cohomological dimension.
