Given some absolute constant $C$ (In my case, $C=4$ would suffice) and an elliptic curve $E/\mathbb{Q}$, are there upper bounds on $|L(E,s)|$ that are uniform for $|s|<C$? Using the functional equation we see that $|L(E,s)|\gg N\_{E/\mathbb{Q}}$ for at least some points $s$, and so would it be possible to prove some sort of result of the type


$$\sup\_{\substack{N\_{E/Q}<B \\ |s|<C}}|L(E,s)|\ll B\tag{1}$$


where the implied constant depends at most on $C$?


The reason I ask this question is that I have a family of elliptic curves $E\_n$, and using some simple facts about reductions mod $p$ I have that $E\_n(\mathbb{F}\_p)=E\_{\infty}(\mathbb{F}\_p)$ for every $p<n$ and some "final" curve $E\_{\infty}$. As a consequence,


$$\lim\_{n\to\infty}L(E\_n,s)=L(E\_{\infty},s)$$


uniformly on $\Re(s)>3/2$. Now, I would like to construct the power series of $L(E\_n,s)$ around, say, $s=7/4$. Because of the convergence of $L(E\_n,s)$ to $L(E\_{\infty},s)$ in $\Re(s)>\frac{3}{2}$ would locally resemble to $L(E\_{\infty},s)$. Moreover, if I had some strong result of the type (1) then I could bound the error and conclude that $L(E\_n,1)$ was similar (converges to) to $L(E\_{\infty},1)$. Since $E\_{\infty}$ has rank 0, we can use Gross-Zagier to conclude that $E\_n$ has rank 0 as well (at least for large enough $n$).


This sort of argument by power series feels super powerful, and it makes me really wonder whether or not bounds like (1) exist. If they don't I will try to find some on my own.



Are there upper bounds on the L function $|L(E,s)|$ for $|s|<C$?
Assuming the modularity theorem, apply the maximum modulus principle to $$\Lambda(E,s)=N^{s/2}(2\pi)^{-s}\Gamma(s)L(E,s)$$ which is entire and $\Lambda(E,s)=\pm \Lambda(E,2-s)$ (where $N$ is the conductor).


The Hasse bound gives a bound for $\log L(E,s)$ on $\Re(s)=C+2$, this gives a bound for $\Lambda(E,s)$ on $\Re(s)=C+2$ and $\Re(s)=-C$, and since it is entire and rapidly decreasing as $|\Im(s)|\to \infty$ this gives a bound for $\Lambda(E,s)$ on $\Re(s)\in [-C,2+C]$ depending only on $N$ and $C$.


Given a sequence of elliptic curves $E\_j$ with conductor $N\_j \to \infty$ then $|L(E\_j,-1/2)|\to \infty$.


