Is this language in P?
Using the product construction, given $M$ you can construct (in linear time) a DFA for the language $L'(M) := L(M) \cap \Sigma(\Sigma\Sigma)^*$. Using DFS/BFS, you can check whether $L'(M)$ is empty or not (you check whether some accepting state is reachable from the initial state) – $L'(M)$ is empty iff $M \in L$. The entire algorithm runs in linear time.