Multiple Dedekind Zeta Functions

Riemann zeta function has a number field analogues, which are called Dedekind zeta functions. The multiple Dedekind zeta functions (MDZF) should be considered as a number filed analogue of multiple zeta function. MDZF have integral representation, infinite sum representation, and analytic continuation.
The integrals used in the definitions are higher dimensional analogue of iterated path integrals, which give functions on higher path spaces. I will give partial answer to some questions such as: Are the values of MDZF periods? Do they satisfy shuffle relations?