The complexity class ModL was defined by Arvind and Vijayaraghavan in [AV04] (more precisely in Definition 1.4.1, Vij08],[Definition 3.1, AV]). In this paper, under the assumption that NL =UL, we show that for every language LModL there exists a function f\sharpL and a function gFL such that on any input string x, we have1. g(x)=0p for some prime p, and,2. if xL then f(x)1(\modp),3. if xL then f(x)0(\modp).

As a consequence under the assumption that NL=UL we show that1. \ModL is the logspace analogue of the complexity class ModP defined by K\"obler and Toda in \cite[Definition 3.1,KT96], and that2. \ModL is closed under complement.

We prove the characterization of ModL stated above by showing the following property of \sharpL. Assuming NL =UL, if f\sharpL and gFL such that g(x) is a positive integer k in unary that depends on the input x then the functionkf(x)\sharpL .