Functional Equation of Zeta Function on Statistical Model

I’ve been studying [1] because I was interested in his ideas on the zeta function. I’ll define it here (c.f. p. 31):

The Kullback-Leibler distance is defined as $ $ K(w)=\int q(x)f(x, w)dx\quad f(x,w)=\log\frac{q(x)}{p(x|w)} $ $

For a given triple $ (p, q, \varphi)$ , where $ p(x|w)$ is a statistical model (a p.d.f. at parameter $ w$ ), $ q(x)$ is a true probability distribution, and $ \varphi(w)$ is an $ \textit{a priori}$ probability density function with compact support, its zeta function for $ z\in\mathbb{C}$ $ $ \zeta(z)=\int K(w)^z\varphi(w)dw $ $ ($ K$ is generalized to any positive analytic function). I believe he is serious that is within the family of zeta functions because he ends the relevant chapter on the derivation and properties of the Riemann zeta function ([1], p. 132). However I understand there to be a criteria for (arithmetic) zeta functions [2]:

  1. Algebraicity $ $ L(s)=\sum a_nn^{-s},\quad a_n\in\mathbb{Z} $ $
  2. Euler product $ $ L(s)=\prod\phi_p(p^{-s})\quad\phi_p(X)\text{ is rational with bounded degree} $ $
  3. Functional Equation: For some $ h>0$ specific to the zeta function, the following holds for $ z\in\mathbb{C}$ : there exists a ‘gamma factor’ $ \gamma(z)$ specific to the L-function $ $ \gamma(h-z)\zeta(h-z)=\gamma(z)\zeta(z) $ $
  4. Special values (see cited paper) $ $ \S$ $

Question 1: In the interest of studying the zeta function of statistical models rigorously, has there been any effort to expand Zagier’s (or similar) criteria to non-arithmetic zeta functions?

Question 2: Is there any research since this publication on the zeta function of statistical models? Particularly, any efforts to establish a functional equation?

$ $ \S$ $

[1]: Watanabe, S. Algebraic Geometry and Statistical Learning Theory

[2]: Zagier, D.