That's what I thought, but as I was pretty sure to have CC displays I thought I'd ask first.
So I'm even more confused now, I managed to get different displays (DC56-11SYKWA) which are CC - and have a proper datasheet ! 
But still the same issue. I rewired everything a few times, checked everything I could, and still pitch black.
And again, switching one of the digit pin with one segment lights it up, so it seems that it is somehow inverted but I can't figure out why.
To make it a bit more clear, I plugged the pin 13 of the display on pin 10, and the segment b went on.
Any idea of what I could've done wrong ? 