PA-23 Flight instructor reported an engine failure while doing air work with a student. Pilot secured engine and landed normally.

Narrative

During a routine training flight my student and I experienced a complete failure of the right engine. While recovering from slow flight back to normal cruise; the right engine began to sputter and eventually died. After running the appropriate checklists and attempting to re-start the engine two (2) times; I requested assistance with ZZZ tower and informed them that we were a light twin who had lost an engine and requested to land. As we were at 6;000 feet 12 miles north of the field; we performed two (2) left 360* turns to lose altitude before performing a straight in landing to Runway XX. The landing itself was uneventful; and we successfully taxied to the ramp for shutdown. We were met by the fire department to debrief; and we also called the Tower on the phone per their request. There was no fire; smoke; or liquids weeping from the engine. There were no injuries to persons or property. ZZZ tower had to vector some traffic to give us priority; and they were otherwise generally helpful and professional.