Zeroes of L-series in characteristic p

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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 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 th2 and quite recently, Javier Diaz-Vargas dv2. In the examples computed it was found that these non-classical trivial zeroes were correlated with integers having 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.