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An E06 error is sometimes shown by an Xbox 360 console (displayed on TV) with a message prompting the user to contact Microsoft Customer Support. In addition to the on-screen error, the console is also usually blinking a single red light in section 4 in the ring of light on the console (as shown in the picture). There is some information floating around about what this error means and what a user can do to fix it.
An E06 error is attributed to an overheating GPU (GPU pic shown below). If you have standard warranty, you should contact Microsoft Customer Support about this problem. It could be caused by a bad connection between the heatsink and the GPU (perhaps the heatsink is not bolted in tightly enough). Another problem is poorly applied thermal paste during the manufacturing of the unit. Thermal paste transfers heat from the chip to the heatsink, but if there is not enough applied or it is not applied evenly, it will fail to cool the chip adequately.
A user with no warranty could opt to disassemble the console (How to disassemble Xbox 360), remove the silver heatsink that cover the GPU (by removing the X-Clamp on the underside of the board), clean the chip and heatsink with Arcticlean or isopropanol alcohol, apply new thermal paste (Arctic Silver 5 is highly recommended - just a small dab a little bigger than a grain of rice at the center of both chips) and then re-attaching the heatsink properly (if attached correctly, the heatsink will spread the compound across the chips itself).
The E06 error is associated with the error code 0012. To get an error code from an Xbox 360 console that is displaying an E06 error (or 3ROL) you simply press and hold the "sync up" button (used to connect wireless controllers) and then press the Eject button for the DVD drive. This will bring up the first digit of the error code (usually it is all four lights blinking red, which represents 0, not 4). With the sync up button still held, press the Eject button again and you get the second digit. Do this twice more to get the full 4 digits of the error code.
For more information on the error code you receive, or on Xbox 360 hardware problems in general, read this Xbox-Scene forum thread.