It's a good place to start. The second light usually indicates that the system is powered on. So first thing's first I would make sure you're trying to power up the system before checking the LEDs.
If this result persists after you have tried turning the machine on it could be absolutely be a faulty power supply. It could also be a faulty power button. But power supply is more likely and is a much easier fix since the power button is typically built into the whole chassis and is a bit a of a bear to replace.