I had the same symptoms with an Olds. Figured out that it was the gas tank filter (mesh sock) at the fuel pump. I had added some strong injector cleaner to the tank and it must have cleaned off the residue inside the tank. The stuff would get sucked up to the fuel pump and clog up the sock filter, cutting off fuel supply. Car would die. Then it would start after setting awhile. The dirt would float away from the sock and allow the car to start. I thought I was fighting a heat/electrical problem. Had to replace filter and fuel pump and clean the tank. I do not know if our Luminas have the "sock" filter in the tank. Good luck.