2005 Chrysler Crossfire Q&A
2005 Chrysler Crossfire Question: why does my car not crank or shuts off while driving?
hard to tell but when you said it wont start after it cuts off then that sounds like the crankshaft position sensor.a good mechanic should be able to diagnose a good or bad ckp sensor. - joelburke26
scan test for codes - goodguy