Unit L-functions and a conjecture of Katz

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Let f: X -> Y be a separated morphism of schemes of finite type over a finite field of characteristic p, let Lambda be an artinian local Zₚ-algebra with finite residue field, let m be the maximal ideal of Lambda, and let L be a bounded constructible complex of sheaves of finite free Lambda-modules on the étale site of Y. We show that the ratio of L-functions L(X,L)/L(Y,f! L), which is a priori an element of 1+T Lambda[[T]], in fact lies in 1+ m T Lambda [T]. This implies a conjecture of Katz predicting the location of the zeroes and poles of the L-function of a p-adic étale lisse sheaf on the closed unit disk in terms of étale cohomology with compact support.