Xbox1 S controller vibrates for 5 seconds and turns itself off right after turning it on.

Unfortunately the only way to turn off vibration on the controller is if a game has that feature. The console itself does not have a way of turning off vibrations, hopefully it will be a future update.

Sounds like the vibrator is going out. Is the controller under warranty? https://support.microsoft.com/en-us/devices

I am not sure if my wired controller is under a warranty. Is there any way I can get information?

All controllers carry a 90 day warranty

Try performing a hard reset EXACTLY like this. Hold the power button on the console for 8 seconds or until the power shuts down completely.
Unplug the power cord from the back of the console for 5 minutes.
Plug the power cord back in and restart the console.