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If with Mesopotamia you mean the ancient civilisations in Babylonia and Assyria before the Persian conquest (that is: before 535 BC), then we need to say that there were no coins at all. In a Mesopotamian context, a shekel is a unit of weight, not a struck coin. The first coins in the world were minted in Lydia around 600 BC. In Babylonia, coins began to circulate during the Persian (Achaemenid) period.
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The only Babylonian coins known and recovered are shekels. The minah was a unit of weight. I have not heard of the gerah being used in a Babylonian context. The Babylonian practice was to put silver pieces in a linen bag and then seal the bag and stamp it with an official weight.