We consider a matrix constructed from a finite set of prime numbers and a real parameter. The largest eigenvalue of this matrix is expressed as half the sum of two quantities: a sum independent of the parameter and the modulus of an exponential sum over primes. We prove two unconditional results:· The logarithm of this eigenvalue is a convex function of the parameter.· A certain symmetric product, formed from this eigenvalue and its value at the symmetric point, attains its global minimum at a remarkable point—the critical half-line.The proofs rest on an exact algebraic identity and the Cauchy-Schwarz inequality. No hypothesis on the zeros of the zeta function is used.This work introduces an original spectral object associated with a finite set of prime numbers. The angular Gram matrix, of rank at most two, encodes correlations between prime powers through oscillating phases. Its variational study reveals convexity and symmetry properties that naturally distinguish the critical line sigma = 1/2. These results, entirely unconditional, offer a new framework for exploring connections between spectral structures and the distribution of prime numbers. They also open perspectives on asymptotic behavior as the set size tends to infinity—a question that remains open.