Quasi-inner functions and local factors

We introduce the notion of quasi-inner function and show that product u = ρ ∞ ∏ v m + 1 ratios local L -factors ( z ) γ / − over a finite set F places Q inclusive archimedean place is on left critical line ℜ 2 in following sense. The off diagonal part 21 matrix multiplication by orthogonal decomposition Hilbert space square integrable functions into Hardy H its complement compact operator. When interpreted unit disk, condition means associated Haenkel compact. none individual non-archimedean and, order to prove our main result we use Gauss theorem factor ratio whose with each retains property be quasi-inner. Finally Sonin's simply kernel 22 for , when ∈ kernels form an inductive system infinite dimensional spaces which are semi-local analogues (classical) spaces.