I am trying to read this paper Poles of Artin $L-$functions and the strong Artin conjecture
On page $1095,$ we are trying to estimate the integral $$\int L(s, \rho) e^{i(\pi/2 - \delta ) s} (2 \pi )^{-s} \Gamma (s - 3/2) E_{-1}(s) ds$$
where $L(s, \rho)$ is Artin $L$-function with 2-dimensional, icosahedral representation $\rho$.
By shifting the contour of the integral to $\mathrm{Re} (s) = 3/2 + \Delta$ with $0 < \Delta < 1/2$, the paper says that the integrand is $\mathrm{O} \left(\frac{1}{|s|^{3/2 - \Delta} } \right)$.
I am trying to figure out how he came to the conclusion that the integrand is $\mathrm{O} \left(\frac{1}{|s|^{3/2 - \Delta} } \right)$.
It is given in Lemma $3$ that $E_{-1}(s)$ is $\mathrm{O} (1/s)$ in $\mathrm{Re} (s) > 1$.
We can use the Stirling formula for the gamma function in the integral above. But I have no idea how to bound the Artin $L$-function in the integral above.
For the integrand to be $\mathrm{O} \left(\frac{1}{|s|^{3/2 - \Delta} } \right),$ it seems like $L(s, \rho) = \mathrm{O} (\frac{1}{s^k})$, $k \in \mathbb{Z}_{>0}$ has to be true. I am not sure whether it is true that $L(s, \rho) = \mathrm{O} (\frac{1}{s^k})$, $k \in \mathbb{Z}_{>0}$ or how to derive this expression even if it's true.
How can I eventually derive the bound $\mathrm{O} (\frac{1}{|s|^{3/2 - \Delta} } )$ for the integrand? I just started reading about Artin $L$-functions, so I might have missed some basic properties.