A political role, rather than a popularity contest.
Hannover is a central location for state politics. The Niedersachsen Landtag works in the Leineschloss. A plenary sitting of Niedersachsen’s Landtag therefore means a meeting of the state parliament in Hannover.
Braunschweig, Wolfsburg and Osnabrück are also cities in Niedersachsen. For this question, placing a city in the right federal state is not enough: only Hannover has the explicitly established capital role.
The other three answers
The other cities are real, but are not the capital asked for.
Braunschweig
Braunschweig is in Niedersachsen, but is not its present-day state capital. Historical importance does not alter that distinction.
Wolfsburg
Wolfsburg is a city in Niedersachsen. Being well known or economically important does not automatically make a city the capital.
Osnabrück
Osnabrück also belongs to Niedersachsen. The constitution, however, names Hannover as state capital, rather than Osnabrück.
Distinguish state politics from city politics.
Alongside its capital role, Hannover has its own municipal administration. A city matter and a decision of the Landtag belong to different political levels, even if both take place in the same city.
A state capital need not lie at the geographical centre of its state. Here the constitutional designation is decisive. A current map is useful for a journey; Hannover is enough for the catalogue answer.
Article 1 · State capital · Constitution of Niedersachsen ↗Niedersachsen → Hannover.
The Landtag in the Leineschloss makes the connection to state politics concrete.
Three words for this question
- die Landeshauptstadtstate capital
- The city designated as a federal state’s capital.
- das Landesparlamentstate parliament
- The elected representative assembly of a federal state.
- die Plenarsitzungplenary sitting
- A sitting of the full parliamentary assembly.