Xbox 360 Controller - Left Thumbstick moves by itself going down and up all the time
I would assume it's most likely an issue with the sensor on the left analogue stick since your right one behaves completely fine. If you still have warranty on it it's best to just get it replaced.
Alternatively what you can try is opening up the controller, cleaning the analogue stick module and watching for loose solder points. Here is a guide that can help you identify the solder points and here are the disassembly steps. Maybe you are lucky enough and a good clean is all it needs.
You can also replace the whole analogue stick unit with a new one if all else fails.