DA42 student pilot reported an engine failed to restart after being shut down. Pilot reported to departure airport and landed safely.

Narrative

During my check ride event on Day 0 the dpe had me shut off the critical engine as part of the exam. I had the QRH out and opened I followed the check list and started a timer. After about 4 minutes flying with one engine inoperative. He told me I could restart the engine. I proceeded by following the unfeather and restarting the engine in flight check list. When I engaged the starter the engine almost started then stopped. I waited then tried to follow the procedure again. This time the engine started to windmilling and wouldn't stop. I had no power in the left engine. He asked what I wanted to do. I told him I wanted to head back to the airport. He agreed. He took over comms and I kept the controls proceeded direct to ZZZ and ran check lists. I secured the engine and ran the landing with one engine inoperative check list. Focused on maintaining a safe airspeed with a stable descent. I landed safely with the critical engine secured and taxi back to the ramp safely. No injuries no damage.