  In the classical theory of $L$-series, the exact order (of zero) at a trivial
zero is easily computed via the functional equation. In the characteristic $p$
theory, it has long been known that a functional equation of classical
$s\mapsto 1-s$ type could not exist. In fact, there exist trivial zeroes whose
order of zero is ``too high;'' we call such trivial zeroes ``non-classical.''
This class of trivial zeroes was originally studied by Dinesh Thakur \cite{th2}
and quite recently, Javier Diaz-Vargas \cite{dv2}. In the examples computed it
was found that these non-classical trivial zeroes were correlated with integers
having {\it bounded} sum of $p$-adic coefficients. In this paper we present a
general conjecture along these lines and explain how this conjecture fits in
with previous work on the zeroes of such characteristic $p$ functions. In
particular, a solution to this conjecture might entail finding the ``correct''
functional equations in finite characteristic.
