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My Sony Xperia Z5 just shut down and cannot turn on anymore, I tried repairing the software using Xperia Companion but after connecting the USB cable and waiting for the application to proceed to the next step, nothing happens
Hi @asavieri, I am sorry to hear about that. 
Xperia Companion can only proceed with the repair if the device is turned off, therefore, could you try forcing a shutdown by pressing and holding down the power- and volume up buttons simultaneously until the device vibrates 3 times? Alternatively, you may press the OFF button located next to the SIM area as @asavieri mentions, wait for 3 consecutive vibrations here as well.
Prior to trying the software repair, try forcing a shutdown of the device as I described above and then try to boot the device, as the case could be that it simply froze and therefore is unresponsive.
You may also try dismounting any SIM and SD card which you may have mounted, and then try booting the device to see if that helps.
In addition, I recommend trying to charge the device for up to 2 hours with an original charger and cable before trying to boot it. If the device had a completely discharged battery, then it may take up to 30 minutes before it start reacting to the charge. During this period the LED may not light up and the screen may remain completely dark.
Let me know how it went! If my post helped, feel free to kudo it and mark it as the accepted solution so that other users facing the same concern can find it! 
I tried everything you mentioned, what actually happened was I was using the phone and it suddenly froze, then the screen went dark, I then proceeded to switch it off using the switch next to the sim card tray. After the 3 vibrations that was it... it never switched on again, when the charger is connected it does not even show the led light, and when connected to the PC it is not being recognized. So when i searched online that's when I found out about the Xperia Companion method and tried it but it never seemed to work.
I am sorry to hear that.
If it still does not turn on even after you tried the troubleshooting that I mentioned then I recommend that you have it sent to an authorized service center so that a technician may have a look at it.
In order to have it sent in, I recommend getting in touch with your Local support as they are well informed regarding your local service routines, warranty terms and consumer laws.