Consider any language L that is undecidable but recursively enumerable. Show that any recognizer for L must loop on infinitely many inputs.

Hint:
Suppose that, M is some DTM that accepts L. Let S be the set of inputs for which M loops forever. Show that if S is finite, then there exists a deterministic TM N_S to decide L (S is hardcoded in N_S). Proceed with the proof from here.