Periodic-Sequence Kernels and Elementary Closed-Form Identities for Gamma-Weighted L^2-Means of Dirichlet L-Functions

Let $a=(a(1),\dots,a(q))$ be a real sequence of period $q$ satisfying the zero-mean condition $\sum_{n=1}^{q}a(n)=0$. Denote by $D_a(s)=\sum_{n\ge1}a(n)n^{-s}$ its Dirichlet series and by $P_a(u)=\sum_{n=1}^{q}a(n)u^n$ the associated polynomial. We prove a general closed-form theorem: the corresponding Gamma-weighted $L^2$-mean$$\frac{1}{2\pi}\int_{-\infty}^{\infty}\bigl|\Gamma(1/2+i\tau)\,D_a(1/2+i\tau)\bigr|^2d\tau$$is exactly equal to the rational-function integral $\int_0^1|P_a(u)|^2/[u|1-u^q|^2]\,du$, and this integral can always be written in the form$$r+\sum_{j=1}^{q-1}c_j\log\!\Bigl(2\sin\frac{\pi j}{q}\Bigr)+r'\pi,$$where the coefficients are algebraic numbers in the real subfield of the cyclotomic field. As the first nontrivial instance of this theorem, we establish the new identity$$\frac{1}{2\pi}\int_{-\infty}^{\infty}\bigl|\Gamma(1/2+i\tau)\,L(1/2+i\tau,\chi_3)\bigr|^2d\tau=\frac{1}{3}-\frac{\pi}{9\sqrt{3}},$$where $\chi_3$ is the nonprincipal Dirichlet character modulo $3$. We further prove that the closed-form property holds precisely for all zero-mean periodic sequences; for nonzero-mean sequences, the kernel must be subtracted by its pole term before it belongs to $L^2(0,\infty)$, and the subtraction term destroys the rationality of the integrand in the logarithmic variable, so that the partial-fraction mechanism fails. This result is complementary to the classical mean-value theory of $L$-functions (Ingham-type unweighted asymptotics): the latter describes the global growth of integrals on the critical line, whereas the present paper gives exact values for Gamma-weighted integrals.