Weakly motivic Shimura's zeta functions via p-adic L-functions and doubling method

A motivic approach is developped to Shimura's zeta functions and
attached $p$-adic $L$-functions via admissible measures and doubling method.
Shimura's zeta functions $\Zc(s,{\f})$ \cite{Shi00} attached to
holomorphic arithmetical automorphic forms ${\f}$} on unitary groups
$U_K$ of an imaginary quadratic field $K$.
A motivically normalized $L$-function $\Dr(s,{\f})$ attached to
$\Zc(s,{\f})$  is defined in accordance with Deligne  \cite{De79} and
Coates-Perrin-Riou conjectures \cite{CoPe}.
%An explicit description of Shimura's $\Gamma$-factors is used.
The attached $p$-adic $L$-functions  of $\Dr(s,{\f})$ are constructed
via {\it admissible measures of Amice-V\'elu}, see also \cite{MTT}.
The $p$-ordinary case was treated  in \cite{EHLS} via algebraic
geometry (method of Katz).

The main result is stated in terms of the Hodge polygon $P_{H}(t):

[0,d]\to {\mathbb R}$ and the Newton polygon $P_N(t)=P_{N,p}(t):

[0,d]\to {\mathbb R}$of the zeta function $\Dr(s,{\f})$ of degree  $d=4n$.

Main theorem gives a $p$-adic analytic interpolation of the $L$
values in the form of certain integrals with respect to Mazur-type
measures, saisfying  Coates-Perrin-Riou conjectures.
Let us call such Euler product weakly motivic at $p$.
Both Rankin-Selberg and doubling methods are used.