  Given $\mathbf{F}$ a number field with ring of integers
$\mathcal{O}_{\mathbf{F}}$ and $\mathfrak{p},\mathfrak{q}$ two squarefree and
coprime ideals of $\mathcal{O}_{\mathbf{F}}$, we prove a reciprocity relation
for the first moment of the triple product $L$-functions
$L(\pi\otimes\pi_1\otimes\pi_2,\frac{1}{2})$ twisted by
$\lambda_\pi(\mathfrak{p})$, where $\pi_1$ and $\pi_2$ are a fixed unitary
automorphic representation of $\mathrm{PGL}_2(\mathbb{A}_{\mathbf{F}})$ with
$\pi_1$ cuspidal and $\pi$ runs through unitary automorphic representations of
conductor dividing $\mathfrak{q}$. The method uses adelic integral
representations of $L$-functions and the symmetric identity is established for
a particular period. Finally, the integral period is connected to the second
moment via Parseval formula.
