P-adic Rankin-Selberg L-functions in universal deformation families and functional equations

We construct a p-adic Rankin-Selberg L-function associated to the product of two families of modular forms, where the first is an ordinary (Hida) family, and the second an arbitrary universal-deformation family (without any ordinarity condition at p). This gives a function on a 4-dimensional base space - strictly larger than the ordinary eigenvariety, which is 3-dimensional in this case. We prove our p-adic L-function interpolates all critical values of the Rankin-Selberg L-functions for the classical specialisations of our family, and derive a functional equation for our p-adic L-function.