I was going through [this](https://www.ams.org/journals/jams/2003-16-01/S0894-0347-02-00410-1/) paper. Corollary 7.3.4 says the $L$-function $L(s,\pi, \rm{sym}^4)$ is holomorphic except possibly at $s=0,1$ and gives a necessary and sufficient condition for it to have a pole at $s=1$ where $\pi$ is a cuspidal representation of $\rm{GL}\_2(\mathbb{A}\_{\mathbb{Q}})$.


But it does not talk about the behaviour of the fucntion at $s=0$. Can we give a necessary and sufficient condition for the function to have a pole at $s=0$?


Also if $\pi$ is not monomial then, $L(s,\pi, \rm{sym}^4)$ does not have a pole at $s=1$. Can we say more? Can we say it is non-zero or zero at $s=1$?


Similarly, by corollary 5.1.7, $L(s,\rm{sym}^2(\pi)\otimes \chi)$ does not have a pole at s=1 iff $\pi$ is not monomial. Can it have a zero there or is it always non-zero at $s=1$ when $\pi$ is not monomial?


Thank you in advance.



Behaviour of a certain $L$ function at $s=1$
Let $n\geq 1$. The $L$-function of an automorphic representation of $\mathrm{GL}(n)$ is either (1) entire, or (2) holomorphic away from a pole of order $\leq n$ at $s=1+i\tau$ for some fixed $\tau\in\mathbb{R}$. Kim (Theorem B) proved that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}(2)$, then $L(s,\pi,\mathrm{Sym}^4)$ is the $L$-function of an automorphic representation of $\mathrm{GL}(5)$. Therefore, there is no pole on the line $\mathrm{Re}(s)=0$.


The automorphy of $L(s,\pi,\mathrm{Sym}^4)$ follows from the automorphy of the exterior square lift from $\mathrm{GL}(4)$ to $\mathrm{GL}(6)$. The cuspidality criterion for this exterior square lift prove [here](https:///math.okstate.edu/people/asgari/Files/cuspidality.pdf) directly determines the order of the pole at $s=1+i\tau$ of $L(s,\pi,\mathrm{Sym}^4)$.


If $\pi$ is an automorphic representation of $\mathrm{GL}(n)$, then $L(1+it,\pi)\neq 0$ for all $t\in\mathbb{R}$. See the appendix to [this paper](https://www.math.univ-paris13.fr/%7Ebrumley/Research_files/HC-Schwartz.pdf) for a much stronger zero-free region.


With some modifications, all of these comments (holomorphic apart from possible pole at $s=1+i\tau$, classification of factorization that determines the order of the pole at $s=1+i\tau$, zero-free region that includes the line $\mathrm{Re}(s)=1$) apply to $L(s,\mathrm{Sym}^2(\pi)\otimes\chi)$. The cuspidality criterion follows from seminal work of Gelbart and Jacquet.


