Normal Functions, Even Theta Characteristics and the Theta Divisor

Let $[C]$ be a general point in the moduli space of curves $M_g$ with $g > 1$. Let $G \subset J(C)$ be a connected compact subgroup of real dimension $1$ of the Jacobian, and let $L$ be an even theta characteristic on $C$. We prove that $\{\zeta \in G \mid H^0(C, L \otimes \zeta) \neq 0\} = \emptyset$ if and only if $L \otimes \zeta_G$ is an even theta characteristic on $C$, where $\zeta_G$ is the unique non-trivial point of $G$ of order two.