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Yep... The lolipop vwersion of Z2 is... kinda laggy and always thinking. Tried to go to Play store and it was launching the store for so long that connection limit appeared. Also Album App and walkman work like the system is trying to find them. When Z2 was launched it was said that's the phone so quick it waits for your reaction... well, now I get the feeling I'm having a 4 years old device, not a quad-core with 3Gb RAM 
You could try to perform a software repair using PC Companion.The process will erase all internal memory.You might need to charge your device for about 1-2 hours before so the device has at least 60% battery.
Make sure the device is not connected to computer or laptop and turned off (sometimes you have to press power button and volume up until the device vibrates 3 times or on some devices you have to press the small button near the SIM tray or on the back of the device).
Open PC companion->support zone->phone/tablet software update->repair my phone/tablet blue highlighted.Follow the steps given,do not connect your phone until prompted to press volume down while plugging USB cable. If PC companion does not start the repair process, you might need to run it in compatibility mode with Windows 7 (in some cases XP).
i think it would be prudent for Sony to make sure that all updates are good they must have many testers worldwide that are independent that can verify that the update in question is fit for purpose. all of the z2 specifications must surely be equal across the world or am i missing something here in which the z2 is not equal
my gut feeling is that the lollipop update has been rushed through as it effects all makes of mobiles with a whole range of problem's ?
I've been there already.. that's my third Clean install of lolipop with whipped out everything stuff, I guess Lolipop needs more time to set himself up than kitkat needed. And charging always has issuess around 83 - 94% it takes hours to charge 5 to 7%, weird cause updating lolipop the was 100% fully charged, now the system doesn't reconize it. From 22% - 79% it charges "normally" as usual.
I havent noticed any slowdown myself so cant help. maybe an app playing up ?
at 1st i felt my mobile after 5.1 is kind of slow. not much noticeable, but in some point its slowing down. battery also draining faster than before.