Katz type $p$-adic $L$-functions for primes $p$ non-split in the $\mathrm{CM}$ field [Katz type p-adic L-functions for primes p non-split in the CM field]

For every triple F, K, p where F is a classical elliptic eigenform, K is a quadratic imaginary field and p is an odd prime integer which is not split in K, we attach a p- adic L- function which interpolates the algebraic parts of the special values of the complex L- func- tions of F twisted by algebraic Hecke characters of K such that the p- part of their conductor is p n , with n large enough (for p >= 5 it suffices n >= 2 ). This construction extends a classical construction of N. Katz for F an Eisenstein series, and of Bertolini-Darmon-Prasanna for F a cuspform when p is split in K. Moreover, we prove a Kronecker limit formula, respectively, p- adic Gross-Zagier formulae, for our newly defined p- adic L- functions.