The government of a state or province is located in the Capital City, which is the seat of power.
The three branches are executive, legislative, and judiciary.
In my state of California, the first two are not surprisingly based in the capital Sacramento, but the latter one, California Supreme Court is based in San Francisco.
I’m curious to see which states have one branch that is NOT in the capital city in terms of U.S states?
For the state Supreme Courts, Wiki lists six states as “main exceptions”, which have it headquartered somewhere other than the capital city, or multiple coequal locations:
I know you asked about U.S. states, not independent countries, but a surprisingly massive part of the German government is located outside the formal capital city, Berlin. This includes, among many other less-known authorities, the country´s top courts, central bank, and many of the government departments, plus countless federal-level authorities that are scattered all over the country. In fact, there still are more federal civil servants working in the country´s former capital, Bonn, than in the current capital, Berlin.
On the “countries” level, a good example is South Africa, which has each of the traditional 3 branches headquartered in a different city, and may be described as having 3 capitals, no real capital or a “defacto” capital in Pretoria (that, however reflects apartheid era usage):
Cape Town: “legislative” capital.
Pretoria: “administrative” or “executive” capital.
Bloemfontein: “judicial” capital.
Since we’ve moved on to national capitals, there’s also Bolivia. Sucre is officially the capital, and houses the Supreme Court, but La Paz is the ‘seat of government’, and contains both the Presidential Palace and the National Congress.
For the state Supreme courts, you might note that the idea in the six exceptions seems to be locating the Supreme Court in a more populous or economically important city, or setting up multiple locations. Presumably to allow easier access for appearances before the court. Capital cities themselves are often NOT located in the most populous or important city, and there is a school of thought that says they should not be so located. You can make an argument that you don’t want your legislature and governor’s office sitting in your major city.