Put a "tester" on the battery when running and you got 11 VDC? What is a tester? A multimeter? Hopefully, as I will attempt to go through a few steps to get you going again and you will need one for some of this...
1. I know, I know, you have a new battery. But new batteries are sometimes defective; especially if not charged correctly initially. Buy a SMALL hydrometer at any autoparts store. It will look like a 3" long eyedropper and be filled with colored balls. It will also have a small piece of hose for connecting to the hydrometer so you can put the hose into the plug hole in each cell and draw up some fluid into the hydrometer. Note that on the packaging of the hydrometer, they give you a guide to how to interpret the number of balls that are floating. This will tell you if you have a dead cell. If you have a dead cell, you likely charged the new battery incorrectly or it was defective. If this is the case, ask how to charge a battery (if you don't know) and replace the battery. If the battery cells are OK, go to step two.
2. Measure battery voltage at idle and at 4K rpm. At idle, it should measure 12.5 VDC and at 4K rpm it should measure about 14.5 VDC. If you are getting LESS at 4K rpm, then you should look at the connections to your stator/alternator and if MORE than 14.5 VDC, look at your rectifier or regulator/rectifier. If you don't have your bike year/model and location in your signature, put it there by going into your profile and editing your signature. There are different charging systems on various KZ1000 models and really helps to know which I am talking about... if the info is already there, thanks and ignore my whining.
If you have MORE than 14.5 VDC at 4K rpm, your rectifier or regulator rectifier is BAD and should be replaced. If less, then first, remove connections to both your regulator/rectifier or the regulator AND rectifier and use contact cleaner on them and dielectic grease and reconnect securely. Retry the voltage test. If this doesn't fix your problem, you will want to look at your stator or alternator or whatever it is called for your model.
3. Last bit... On most models, the wires that come from the stator or generator or whatever it is called are all yellow and go up and terminate in a BLUE plug at the bike's junction panel. These can get to be in bad shape. You can first remove the blue plug and clean the contacts with contact cleaner and put some dielectic grease on the contacts. Check to see what the voltages are at the battery again at both idle and 4K rpm. They should be in the ranges already noted. If not, then it is likely you have issues with the stator. If this is the case, you can check voltages for your model using a Factory Service Manual... I think there should be between 20-30VAC (AC current here... change multimeter scale) on each leg but use your manual to be sure. If the rectifier or regulator/rectifier aren't getting a full does of AC, the battery won't charge. If you find the stator doesn't check out, drain your oil and remove the side cover. There are three socket head bolts holding the stator in place on most models. Remove these and tap the stator on a bench as the stator will be held in place magnetically and carefully examine the three wires that are soldered to the windings once you get the stator out. If your bike uses a different kind of generator/rotor set up, look in your Factory Service Manual for electrical checks for those components. If you have a stator type set up, I would replace it... they are on eBay all the time.