DUBAI—Saudi authorities are seeking the death penalty for three prominent clerics, rights activists and a government official said, testing the unwritten code that has kept the kingdom’s rulers in power.

Crown Prince Mohammed bin Salman and his father, King Salman, have jailed activists, businessmen and government officials as part of their efforts to reshape Saudi society and economics. But Saudi clerics have long been a power unto themselves, with fame and influence beyond that of others caught up in recent crackdowns.

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