The battery is most probably defect. and then,The manual says:
Battery is exhausted.
When connecting to an electrical outlet using a Charging AC Adapter, the camera
cannot be turned on.
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Confirm all connections.
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Off is selected for Charge by computer in the setup menu.
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When charging by connecting the camera to a computer, battery charging stops when the camera is turned off.
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When charging by connecting the camera to a computer, battery charging stops if the
computer is suspended (sleeping), and the
camera may turn off.
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Depending upon computer specifications,
settings, and status, it may be impossible to
charge the battery inserted into the camera
via computer connection.
So even when you think the battery is charged, it still can be empty. When a rechargeable battery is old, or not often used, it can lose its capacity, not being able to power the equipment it was meant for correctly.To make sure your battery is good, you could try to check the voltage with a multi meter. The voltage should be slightly higher than the 3,6 Volt that is printed on the battery. like 3,8 Volt, when measured. if it is just a fraction lower than the 3,6 is is not fully charged and it will be much lower when put in a camera, after switching on the camera. When you measure 2,4 Volt, 1,2 Volt or even nothing you can be sure the battery is dead.Best option then, would be to order a new battery. But if you know someone with a camera that has the same battery that works, ask if you can check your camera still works with a battery that is ok. because when in a rear situation it is you camera that is defective, you can save the mont for a new battery and look out for a new camera, because the batteries are not cheap.