I've moved this to the FZ6 Fazer Forum as it's more appropriate.
First question: Can you push start the bike? (It may take a few attempts because the failed starts might have left unburned fuel in the system).
If so, the problem may well be with the alternator or regulator/ rectifier meaning the battery is not getting charged properly.
Have a look here for the FZ6 manual with details of how to use a multimeter to check the outputs and see if they're in spec.
https://foc-u.co.uk/index.php?action=downloads;cat=3Also get a Drop Tester (or a friendly local garage) to check the battery as "new" batteries can be faulty (trust me, I know!)