The second moment of $GL(4) \times GL(2)$ $L$-functions at special points

In this paper, we obtain upper bounds for the second moment of $L(u_j \times \phi, \frac{1}{2} + it_j)$, where $\phi$ is a Hecke Maass form for $SL(4, \mathbb Z)$, and $u_j$ is taken from an orthonormal basis of Hecke-Maass forms on $SL(2, \mathbb{Z})$ with eigenvalue $1/4 + t_j^2$. The bounds are consistent with the Lindel\"{o}f hypothesis. Previously these types of upper bounds are available for only $GL(n) \times GL(2)$, where $n \leq 3.$