Suppose I have a set $S$ of $n$ integers picked uniformly at random from $[-L, L].$ Let $z(S)$ be the number of subsets of $S$ which sum to zero. The question is: what is the distributon of the variable $z(S)$ (I am interested in $n$ fixed and $L$ growing)?

Suppose I have a set $S$ of $n$ integers picked independently, uniformly at random from $[-L, L].$ Let $z(S)$ be the number of subsets of $S$ which sum to zero. The question is: what is the distributon of the variable $z(S)$ (I am interested in $n$ fixed and $L$ growing)?