I was reading a paper on Selberg's central limit theorem for the classical automorphic $L$ functions attached to primitive holomorphic cusp form $f$. I can not understand the following equation.
\begin{align*} &\int_{T}^{2T} L(f,\sigma+it)M(f,\sigma+it) dt\\ =& \sum_{n\leq T}\frac{\lambda_f(n)}{n^{\sigma}}\sum_{m<T^{\epsilon}}\frac{a(m)\mu(m)\lambda_f(m)}{m^{\sigma}} \int_{T}^{2T} (mn)^{-it} dt +O(T^{1/2+\epsilon}) \\ =& T+O(T^{1/2+\epsilon}) \end{align*}
Why does the second line of the equation hold from the first line using the approximate functional equation for the $L$ functions?
The Dirichlet series $M(f,\sigma+it)$ is given by $M(f,\sigma+it)=\sum_{n}\frac{\mu(n)a(n)\lambda_f(n)}{n^s}$, where \begin{align*} a(n)= \begin{cases} 1 &\text{if } n<T^{\epsilon} \\ 0 &\text{otherwise}. \end{cases} \end{align*}
Thanks in advance!