A uniqueness theorem for functions in the extended Selberg class

We study the uniqueness of functions in the extended Selberg class. It was shown in Ki (Adv Math 231, 2484–2490, 2012 ) that if for a nonzero complex number c the inverse images L 1 - 1 ( c ) and L 2 - 1 ( c ) of two functions satisfying the same functional equation in the extended Selberg class are the same, then L 1 ( s ) and L 2 ( s ) are identical. Here we prove that this holds even without the assumption that they satisfy the same functional equation.