I probably have the trophy for stupid questions asked on here, but Ill ask em anyway...

I did ALOT of searching beforehand too...

In the upper left corner of the dash, there is a red LED in a black circular plastic housing, mine doesn't blink, never has...

I thought it was because I don't have an alarm, just the factory built-in killing devices.

But I saw another Z that had the same package as mine and his blinks, and all he has is the plain ol keyfob like mine and he can open his doors after locking the car and nothing goes off....(you do it from inside if youre wondering)

 

i recall mine has gone off. just pluck the led out, and double press the lock on your remote and see if it blinks. mines always does its own countdown.
i modded my led with a small piezo buzzer, you can use a VOM (Volt ohm meter) to see if there is voltage going there and perhaps the led burned out?
otherwise if you have a warranty, dont mess with it and take it in for repair.

 

I been trying to search for a HOW-TO, I think if I can find a Blue LED that uses the save voltage/amps I can replace it...