"Does Esperanto have its own culture?"
Translation:Ĉu Esperanto havas propran kulturon?
That sounds as if you know that Esperanto has its own culture and you are asking whether it forgot it at home or whether it has it with it right now.
What you really want to know is whether it has a culture of its own, rather than simply whether it has its culture.
Ne, sed Esperantistoj havas propran kulturon.
"Ne lingvoj sed homoj havas kulturon" -La Esperanta Teamo de Duolingo
No, unless you mean to ask whether Esperanto has the culture belonging to something else, e.g. "Does Esperanto have English's own culture?".
sian propran kulturon might work, though.
Ah, you are right, of course --I had forgotten the reflexive pronoun "sia" is used with "ĝi" as well as "li," "ŝi" and "ili."
So, if "sian propran kulturon" and "propran kulturon" are both correct, is one more common / favored than the other?
I would be inclined to say
- ...havas sian propran kulturon? = ...have its own culture?
- ...havas propran kulturon? = ...have a culture of its own?
That is, the first sentence pair implies that one single culture of its own exists and asks whether Esperanto has it.
The second one does not imply that this culture exists, and so I think it's a better way to ask about it.
For example, I would ask "Do you have brothers or sisters?" rather than "Do you have your brothers and sisters?", or "Does Mary have brown hair?" rather than "Does Mary have her brown hair?".
That said, in English "its own culture" seems reasonable to me even without this necessary implication -- perhaps because of the way "own" works. ("Do you have your own room?" and "Do you have a room of your own?" both sound fine to me while "Do you have your room?" does not.)
I don't know how the majority of fluent Esperanto speakers treat propran but I think the version without sian sounds better if you are asking about the existence rather than presupposing that it already exists.