It is weighted slightly off center to make up for the coriolis
acceleration due the rotation of the earth.

The weight makes it precess as required to appear to
be level.

And the erection mechanism is as others have pointed out...
takes the average downward vector and calls that level.

Bill Hale


wrote:
Every time I get in the aircraft and before I start the engine I notice
the attitude indicator is displaying a nose down and a left turn.

You just described the expected behavior. Why is this surprising to
you?

Are you expecting the AI to jump to level when you turn the master on?
-harry