Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients.


1. Are there any lower bounds known for $\sum\_{p\leq x}|A(1,p)|^2$ or $\sum\_{n\leq x}|A(1,n)|^2$ ? (we know the lower bound $\sum\_{m^2n\leq x}|A(m,n)|^2\gg\_{\delta} x^{1-\delta}$
2. Is something known about the Dirichlet series (functional equation, meromorphicity etc.)
$$\sum\_{n=1}^{\infty}\frac{|A(1,n)|^2}{n^s}?$$


I know that the $GL(2)$ analogues for both questions are well known.



Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms
For $(1)$, I believe the size would still be $\gg x^{1-\epsilon}$. However, if you allow $$\sum\_{p\sim x} |A(p,1)|^2+A|(p^2,1)|^2$$ a lower bound of similar sort is obtained by [Blomer-Maga's paper](https://academic.oup.com/imrn/article/2015/14/5311/779623) Corollary $4.3$. (In any case, one elementary way to know the average size of the Hecke eigenvalues is by writing them in Satake parameters using Shntani's formula.)


For $(2)$, the Dirichlet series appears in the Rankin-Selberg convolution of $$\langle \mathbb{P}f,\overline{\mathbb{P}f}\rangle,$$
where $\mathbb{P}$ is the standard projection operator from $\mathrm{GL}\_3$ to $\mathrm{GL}\_1$ which comes in the definition of global zeta integral. For instance, see chapter $10$ of Goldfeld's automorphic form book. Thus one can obtain functional equation and meromorphic continuation from the above.


