I read some computation theory lecture notes and after citing and proving the proposition: $\emptyset \in S \Rightarrow L_S = \{\langle M \rangle : L(M)\in S\} \notin RE$ it says that $\emptyset\in S$ is not a sufficient condition, i.e $L_S \notin RE$ doesn't yield $\emptyset \in S$, by giving the counter example $L_{\Sigma ^*}:= \{\langle M\rangle : L(M) = \Sigma^*\}\notin RE$ . However it says that there is a necessary and sufficient condition under which $L_S\notin RE$. I searched in Sipser's book for this condition but couldn't find it. I would really appreciate a reference for this condition.
Edit : given the answer by @dkaeae , I wish to know what is the stronger property one can deduce about a non-trivial $S\subset RE$ in case $L_S \notin RE$.