Are you using the choke? GZ's are cold blooded and need the choke when cold even in warm climates.
The GZ's idle speed is 1200 to 1500 rpm. Most people have theirs set too low and that will result in stalling especially when cold as well as oil starvation at an idle.
Have you checked the vacuum hose connected to the back of the petcock? That hose allows gas to flow. It's essentially a vacuum operated shutoff. It must be in place and free of cracks and/or leaks or else the carb bowl will empty out after you prime it and it will stop running. This one is a must!!
Add some in tank carb cleaner (Berrymans or Seafoam). If you can get her to run (even having to rev it to keep it going) take it out and run the crap out of her keeping the revs high to move a lot of treated gas through the carb.
If none of the above works, I suggest you take it to a mechanic or the local stealer.
