Bound on Von Mangoldt for automorphic L-functions

Question

Following the notation in Iwaniec+Kowalski, let $L(f,s)$ be an L-function. Denote $$\frac{L'}{L}(f,s)=\sum_{n\ge1} \Lambda_f(n)n^{-s} $$ In terms of the local roots of the Euler product: $$ \Lambda_f(p^k)=\sum_{j=1}^d \alpha_j(p)^k\log p$$ where $d$ is the degree of the L-function.

QUESTION: I am interested in (the name of, and) any known general bounds for the $\sum_{j=1}^d \alpha_j(p)^k$ term for automorphic L-functions or the entire Selberg class.

Rudnick and Sarnak have a paper where they give the result (due to Jacquet and Shalika): $$ |\alpha_j(p)^k|<p^{1/2}$$ but i believe this citation is only for cuspidal functions.

I would appreciate any guidance or citable references. thanks!

Answer 1

Luo, Rudnick, and Sarnak prove that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$, then $$ |\alpha_{j,\pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{n^2+1}} $$ for all $1\leq j\leq n$ and any unramified prime $p$. The ramification restriction was removed by Blomer and Brumley.

For $1\leq k\leq K$, let $\pi_k$ be a cuspidal automorphic representation of $\mathrm{GL}_{n_k}(\mathbb{A}_{\mathbb{Q}})$. Consider the isobaric sum $\Pi = \pi_1\boxplus \pi_2 \boxplus \cdots\boxplus \pi_K$. Then $$ L(s,\Pi) = \prod_{k=1}^K L(s,\pi_k), $$ and a bound for the local roots of $\Pi$ at $p$ reduces to a combination of the bounds for each of the cuspidal constituents $\pi_k$. Here is a convenient and uniform (yet sub-optimal) bound: If $N$ is the maximum of the $n_k$'s, then $$ |\alpha_{j,\Pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{N^2+1}},\quad 1\leq j\leq n_1+\cdots+n_K. $$ If $\pi$ (resp. $\pi'$) is a cuspidal automorphic representation of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$ (resp. $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$), then one has $$ |\alpha_{j,j',\pi\times\pi'}(p)|\leq p^{\frac{1}{2}-\frac{1}{m^2+1}+\frac{1}{2}-\frac{1}{n^2+1}}=p^{1-\frac{1}{m^2+1}-\frac{1}{n^2+1}} $$ where $1\leq j\leq m$ and $1\leq j'\leq n$.

This follows from the explicit description of the local roots (including at the ramified primes) given by Brumley in the appendix to this paper.

ADDED: In the Selberg class, if you strictly adhere to the axioms, there is no notion of $\alpha_{j,\pi}(p)$. One has a Dirichlet series for $L(s)$ and for $\log L(s)$, but you are not guaranteed that the Dirichlet series for $L(s)$ factors as a product of local roots like $$ \displaystyle \prod_{p}\prod_{j=1}^m(1-\alpha_{j,\pi}(p)p^{-s})^{-1}. $$