[mrlmd1]

Get a battery charger, charge it up completely and then see what happens. There is nothing wrong with your starter, you have a dead battery. After you charge it up fully, let it sit for a half hour or so and then measure the battery voltage. A fully charged battery should be at 12.6-12,8 V. If it doesn't hold that voltage, charge it up again, bring it to a garage or battery place and ask them to do a load test. That's the only way to tell if a battery is any good.It will be free because they want to sell you a new one. If you do buy a new battery, and practically all of them will be the AGM type for the motorcycle, regardless of what the guy in the battery shop says to you, you MUST CHARGE IT FULLY BEFORE FIRST USE or the battery will very quickly fail after only a few starts of the bike and you will be back in the shop buying another one.
And do not start the bike and let it idle thinking you are charging up the battery, Sitting for long periods idling is bad for the bike in many ways, and you will use more "juice" to start it than you will replace by idling. You need to go for a half hour ride at at least 2500-3500 rpms to get anything out of the GZ's charging system. If the bike will be sitting for a few weeks without being ridden in the cold, hook it up to the 1.5-2 amp trickle charger every week or so and you will not have any problem with it. These chargers come with pigtails that you leave connected to the battery terminals so that all you have to do is plug the 2 wires together and plug the charger into the wall. A very simple solution to your problem.