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We study the condition number of the stiffness matrix arising from finite element discretizations of nonlocal integral operators with singular and integrable kernels. Such operators are used in nonlocal diffusion, peridynamics formulation of continuum mechanics, image processing, and phase transition. We quantify of the extremal eigenvalues with respect to all underlying parameters; the horizon, the mesh size, and regularity of the fractional Sobolev space. We prove the sharpness of the proposed quantifications and support the sharpness result with numerical experiments.