[mrlmd1]

Get or borrow a multimeter.

Start up the bike with the battery in it and test the voltage at the battery terminals with the engine running at at least 1/2 throttle (in neutral) and you can see the amount of voltage the charging system is putting out.

Your battery may still read full charge at 13V at rest but have little capacity.That's what a load test measures. Measure the voltage while trying to start the bike and immediately afterwards and if it reads 7-8V you know it is shot.
A bad battery on the bike will never really get charged sufficiently to top it up enough to restart the bike after you draw it down (by starting it the first time) because the charging system of the GZ is so puny.

If you wind up buying a new battery, and most of these are now the sealed AGM type, regardless of what the store owner tells you that it's ready to go after they fill it, put the battery on a charger for a few hours or overnight and make sure it is FULLY charged otherwise it will fail very prematurely and never charge up to it's capacity. Even if it says 12.8 V when you test it out of the box or in the store, it is NOT fully charged. You will get one or two starts out of it and have to return it for another one and keep thinking something is wrong with the bike. Be warned - charge the new battery fully before you use it.

There are numerous threads on here about batteries, I even started one many months ago and then became a semi-expert trying to solve my own problem.