Regarding all particles at a fixed site as a cluster, the size of the largest cluster under the zero range invariant measures is well studied by Jeon et al.[5] for the case of density one. Here, the density of the finite zero-range process is given by the ratio between the number m of particles and the number n of sites. In this paper, we study the lower density case, i.e., the case m = o(n). Especially, when m ~ <TEX>$n^{\beta}$</TEX>,0 < <TEX>${\beta}$</TEX> < 1, we show that there is an interesting cutoff point around <TEX>$\beta$</TEX> = 1/2.