In my previous case,i also faced problem like you,except that its yellow in colour at Z1.
What i do was,install AnTuTu benchmark and run "Stability Test"
It will run test on various temperatures and resulting your phone to overheat.
However i'm not sure whether this could solve it or not.
But its worth the shot eh?
If everything doesn't solve it,i think you are left to send your phone to a local Sony Xperia Center for repair.