Why is my psp go completely unresponsive?

I just got a used psp go N1001 but it didn't have a charger so I ordered one off Ebay, however i'm getting nothing when I plug it in. No charging led or anything. I have no idea if the charger is broken or if the system needs a new battery or something. I don't have another charger so I can't really tell if the charger is working or not. I tried plugging it into a USB port but it is still completely unresponsive. The power brick is an official Sony one but the USB cable is not. Any ideas? Can they run off ac power alone? I have a crappy multimeter but I have no idea how to check the cable with it.