Explicit Formula for the Symmetric Coherence Kernel Associated with Dirichlet L-Functions

We introduce the symmetric coherence kernel K_χ(σ, t) associated with a primitive Dirichlet character χ modulo q. The kernel is defined via an absolutely convergent integral of the character theta function weighted by 1/cosh(u/2). The main result is Theorem 3.1, which provides an explicit formula for the kernel on the critical line σ = 1/2: K_χ(1/2, t) = (τ(χ)/√q) · (π/cosh(πt)) · Λ(1/2 + it, χ), where τ(χ) is the Gauss sum, and Λ(s, χ) = (q/π)^(s/2) Γ(s/2) L(s, χ) is the completed Dirichlet L-function. The proof employs the Mellin transform, contour shift, residue computation, and the functional equation of the L-function. A corollary establishes that the squared modulus of the kernel is proportional to |L(1/2+it, χ)|² with a positive factor, so that the zeros of the kernel coincide exactly with the critical zeros of the L-function.