If you build the culture starbase in the same sector as the planet you want to convert, you get 5x the effect compared to building it in an adjacent sector. See the current version of the user manual or the encyclopedia for details. However, the AI may object to you building a culture starbase in their sector. Unless you have a clear military lead, you're risking a war.
If you build the culture starbase in an adjacent sector, the AI doesn't seem to mind. You only get 1/5 the effect. However, the adjacent sector may not be such a bad idea anyway if you already have a planet there; you can put your +production upgrades on the same starbase. Plus, the adjacent sector may cover more planets.
In the diagram below, let's use A for a sector with an Altarian planet and T for a sector with a Terran planet, and '.' for an unoccupied sector.
.AA
ATA
A..
If you build a starbase in the middle sector, it will help you convert all 5 As. (suppose you got the T system from conquering a minor) If you build a starbase in the sector to the left of the T, it will really help convert that sector, but it won't help with the two As on the right. Plus, it will lower relations with the Altarians.
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