In the context of Vijayanagar rule, ritual kinship played a significant role. The Vijayanagar rulers considered themselves as rulers on behalf of a specific shrine. Among the options given, the correct answer is option 3, Virupaksha.

Virupaksha is a prominent temple located in Hampi, which was the capital of the Vijayanagara Empire. The temple is dedicated to Lord Shiva and is one of the oldest functioning temples in India. The Vijayanagar rulers claimed to rule on behalf of this shrine, signifying their close association with the deity and asserting their divine authority.

While the other options are also significant in the religious and cultural history of the Vijayanagara Empire, they do not represent the specific shrine that the rulers claimed to be ruling on behalf of. Therefore, option 3, Virupaksha, is the correct answer.

Note: The provided answer is generally correct. However, it`s important to mention that the Vijayanagar rulers also had associations with other shrines, including Vithala, Tirupati, Mallikarjuna, and many more. These shrines held religious and cultural importance, but the hallmark of their rule was specifically associated with the shrine of Virupaksha.