Hellenistic Babylonian Empire

Any idea what the culture/military/architecture of a Hellenistic Babylonian Empire would look like? Basic background would be sometime during Alexander's era a native Babylonian Kingdom rises up and gets forged. It obviously is susceptible to Greek cultural influence, but not too much given their already rich Chaldean culture.

So what kind of cultural influences could you see the Greeks possibly having on this state?
 
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