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Does the noise only happen while moving? If it only happens while moving then it could be coming from a wheel bearing or CV shaft. These would be described more as under the vehicle though.
If it doesn't have to be moving for it to make the noise then I would have a helper inside while you're under the hood trying to locate the source of the noise or at least the area it's coming from. The helper can turn the wheel back and forth as needed and can listen while you move or stabilize components, etc. Sometimes a/c or power steering lines can touch body or other components and rattle, you may be able to hold or push on lines to stop the rattle.
Since the noise stops when turning to the right I would pay most attention to the power steering lines and anything in close vicinity to the steering rack and tie rod ends.
Also watch the belt to make sure what you're hearing isn't belt flutter, The belt may be bouncing around due to a bad tensioner, failed belt, or odd power steering pump loading due to a pump or rack issue. If a belt flutters it can create a noise that sounds like a rattle.