Are you on Windows?
You could go into your control panel in windows, game controllers, and it let's you do a little test to see buttons light up.
I think I remember some controllers think the "D pad" is almost like an analogue stick or a trigger with sensitivity, even though you'd expect it to behave like a regular button.
Good idea, hadn't thought to look on devices. But I have to insert some screenshots, I don't understand well. I'm Italian and the screenshot is in my language.
The box with "Asse X" written underneath represents the left analog stick. The 2 colored bars with the words "Asse Z, rotazione Z" represent the right analog stick. (sounds weird to me, but I get it). As for the buttons, they all work except the select, and the dots all can light up except 11 and 12.
The strange thing (for me) is where it says "Punto di vista", here: that represents the d-pad, and it works. (Arrows appear.)
What should I do? it says that perhaps calibration is necessary, but it only concerns the analog sticks.