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Darmon, Dasgupta and Pollack in 2011 applied the Eisenstein congruence for Hilbert modular forms to prove the rank one Gross conjecture for Deligne-Ribet p-adic L-functions under some technical assumptions. These assumptions were later lifted by Ventullo. In this talk, we will apply their ideas in the setting of CM congruence to compute the first derivative of the Katz p-adic L-functions associated with ring class characters of imaginary quadratic fields at the exceptional zero. We will present a precise first derivate formula of the Katz p-adic L-functions in terms of certain Gross regulator and p-adic logarithms of elliptic units. This proves a formula proposed in a recent work of Betina and Dimitrov. This talk is based on a joint work with Masataka Chida.