I have a homework question about the properties (decidability, Turing-recognizability, etc.) of the language $$ L = \{ \langle M \rangle | \text{$M$ is a TM and $M$ accepts some string $w$ which has 101 as a prefix} \}. $$I have made an attempt at showing decidability of $L$:On input $\langle M, w\rangle$ (where $M$ is a TM and $w \in \sigma^*$):Simulate $M$ on $w$.If $M$ rejects and halts, reject. If $M$ accepts and halts, accept.However, I'm not sure about moving forward after this. I do not want a solution, but I want some ideas/techniques as to what else I can prove about $L$. 
The language of TMs accepting some word starting with 101
complexity theory;formal languages;turing machines;automata