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For every triple F,K,p where F is a classical elliptic eigenform, K is aquadratic imaginary field and p> 3 is a prime integer which is not split in K,we attach a p-adic L function which interpolates the algebraic parts of thespecial values of the complex L-functions of F twisted by certain algebraicHecke characters of K. This construction extends a classical construction of N.Katz, for F an Eisenstein series and of Bertolini-Darmon-Prasana, for F acuspform, when p is split in K.  Moreover we prove a Kronecker limit formula, respectively p-adic Gross-Zagierformulae for our newly defined p-adic L-functions.

Katz type p-adic L-functions for primes p non-split in the CM field