Is my PS3 controller working properly?

I recently ordered a Dualshock 3 off of Ebay, and when I opened it up, I noticed how the red light that usually appeared when the controller was connected was not illuminated. Further more, even though I had been charging it for about an hour or so, it disconnected every time I unplugged my USB cord. Is this the usual behavior of the controllers, or have I ordered a defect by mistake?