Functional equation for LC-functions with even or odd modulator

In a recent work, we introduced $\mathit{\text{LC-functions}}$ $L(s,f)$, associated to a certain real-analytic function $f$ at $0$, extending the concept of the Hurwitz zeta function and its formula. In this paper, we establish the existence of a functional equation for a specific class of LC-functions. More precisely, we demonstrate that if the function $p_f(t):=f(t)(e^t-1)\mathrm{/}t$, called the $\mathit{\text{modulator}}$ of $L(s,f)$, exhibits even or odd symmetry, the $\mathit{\text{LC-function}}\mathit{\text{formula}}$ – a generalization of the Hurwitz formula – naturally simplifies to a functional equation analogous to that of the Dirichlet L-function $L(s,\chi )$, associated to a primitive character $\chi$ of inherent parity. Furthermore, using this equation, we derive a general formula for the values of these LC-functions at even or odd positive integers, depending on whether the modulator $p_f$ is even or odd, respectively. Two illustrative examples of the functional equation are provided for distinct parity of modulators.