PA-44 flight instructor reported they could not restart the engine after shutting it down for training and diverted to the nearest airport.

Narrative

Aircraft X departed ZZZ for a routine CMEL training mission with one MEI and one CMEL student candidate onboard. IP (Flight Instructor) initiated an engine failure in flight emergency procedure in accordance with (IAW) local training practices by selecting the fuel mixture to idle cutoff. Student Pilot (SP) executed all memory items IAW established checklists. SP continued troubleshooting steps and completed all checklist items up to and including 'starter engage.' #1 engine did not turn over. IP (Flight Instructor) reviewed that all checklist items were completed and attempted the air start himself to ensure that the starter switch was fully engaged and to confirm it was not SP error. Crew conducted three attempts in route to home base and contacted ZZZ TRACON for monitoring and sequencing. IP requested priority handling with ZZZ TRACON. After further evaluation of the situation; IP elected to divert to ZZZ1 as it was approximately 10nm closer than the originally intended destination. Upon landing and taxing clear of the runway; IP shut down the aircraft and was towed to the ramp for further trouble shooting.