Alarm acting up

It's a 90 300zx, it doesn't have delayed dome light. Close the door, the light turns off. I connected the door trigger directly to the pin switch at the door.  The alarm turns the dome light on when disarmed, just like it's supposed to.

There is no hood trigger connected to the new alarm, I got lazy & let the original factory alarm keep an eye on the hood. The new alarm arms & disarms the factory alarm.

Occasionally when the car won't start on the first turn of the key, on the second turn, as the car starts, the alarm will let out a little chirp.

I havn't messed with it yet, but this is a new development, as everything was normal right after the install.

I suspect the alarm is trying to tell me something, but the manual doesn't cover anything like this.

Thanks!