  Partial zeta functions of algebraic varieties over finite fields generalize
the classical zeta function by allowing each variable to be defined over a
possibly different extension field of a fixed finite field. Due to this extra
variation their rationality is surprising, and even simple examples are
delicate to compute. For instance, we give a detailed description of the
partial zeta function of an affine curve where the number of unit poles varies,
a property different from classical zeta functions. On the other hand, they do
retain some properties similar to the classical case. To this end, we give
Chevalley-Warning type bounds for partial zeta functions and L-functions
associated to partial exponential sums.
