If you wish to spend more than 90 days in Turkey, you will need to apply for a residency visa. This can be managed locally in Turkey.
There is no specific minimum investment required, though you do need to demonstrate your ability to financially support yourself during your stay, as well as provide an address where you will be based during the period you are resident in Turkey.
You will need to have comprehensive health insurance during your stay. The residence permit is issued for an initial period of 2 years, renewable. After 5 years of legal residency, you may apply for Turkish citizenship