How to come up with a language that is recognizable but not co-recognizable?

Question

Forming a language that is recognizable but not co-recognizable. I'm having trouble coming up with a language with these properties. A recognizable language is a language $A \subseteq \Sigma^*$ iff $A = L(M)$ for some Turing machine $M$.

Co-recognizable is the exact same except the complement of $A$ has to be recognizable.

My question is how do I come up with a language that I know will be recognizable but not co-recognizable?