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We study orthogonal polynomials with periodically modulated recurrencecoefficients when $0$ lies on the hard edge of the spectrum of thecorresponding periodic Jacobi matrix. In particular, we show that theirorthogonality measure is purely absolutely continuous on a real half-line andpurely discrete on its complement. Additionally, we provide the constructiveformula for the density in terms of Tur\'an determinants. Moreover, wedetermine the exact asymptotic behavior of the orthogonal polynomials. Finally,we study scaling limits of the Christoffel-Darboux kernel.

Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case