My D620 has been doing odd things over the past month or so (screen going blank), but it was always fixed with a reboot. However last week the screen went blank and would not recover. On boot up the power light would start, the hard disk light would flash and a second later it would stop, no POST screen, no graphics nothing.
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From your description alone it's just not possible to pin down what the exact problem. My only guess would be some critical motherboard error. It could be CPU, RAM, graphic chip or any other key motherboard component.