Simplest nontrivial example of an L-function yielding information about a Diophantine equation

I got excited while reading Langlands' essay, REPRESENTATION THEORY: ITS RISE AND ITS ROLE IN NUMBER THEORY, because he appears to provide concrete motivation for the study of L-functions:

We have not yet linked diophantine equations to zeta-functions, but when we do, we shall see that as a result of a circle of conjectures and theorems the use of zeta-functions offers a way of deciding whether a given diophantine equation has a solution that can be thousands of times more effective than even sophisticated searches for it. For this, however, it is necessary to be able to calculate their values at certain points with precision, although it need not be very great.

He then gives an example of an $L$-function attached to the equation $x^2+1 = 0$, but not quite an example illustrating the point he made above. He uses the fact that there is a 'congruence condition' that determines how $x^2+1$ factors over finite fields of prime order, to argue that the associated L-function can be defined with unexpectedly large domain:

Because it is so important for the subsequent discussion, I repeat that the function $L(s)$ is originally defined by a product of factors determined according to a diophantine alternative, simple though it be. At first glance there is no reason to think that the function has a meaning outside the region, $s > 1$, where the product obviously converges, but thanks to the regularity that we have observed, the product can be converted to a series that is defined by a periodic function and that can be put in a form that has a meaning for any $s$.

So it appears, at this point in the text, that he has used a certain regularity to argue that the $L$-function has surprisingly large domain. Furthermore he argues that the $L$-function can be computed in practice at points in this domain. So he used information about the Diophantine equation to prove something about the $L$-function. But I am interested to understand how to use $L$-functions to tell me something about a Diophantine equation that I wouldn't know otherwise.

Can someone describe one of the simplest nontrivial examples of the phenomena Langlands refers to in bold above? Does it occur in the case of polynomials in one variable, or is he referring to elliptic curves? I am curious to see an example from the perspective of an 'end-user' who simply has a Diophantine equation he/she wants to get information about, and has access to Sage.

Here is an example of what I'm looking for in a potentially different context. Suppose someone said:

If the BSD (Birch and Swinnerton-Dyer) Conjecture were true, it would allow mathematicians to quickly determine whether certain elliptic curves have a finite or infinite number of solutions!

Then I would hope for an explanation like this:

For example, suppose you wish to understand the rational solutions to $E: y^2=x^3+1$. This equation has a function associated with it, defined on the complex numbers, denoted $L(E,s)$. This function can be evaluated quickly in practice. The BSD conjecture states that the rank of $E(\mathbb{Q})$ is equal to the order of the zero at $s=1$ of the function $L(E,s)$. Go to Sage and write:

E = EllipticCurve([0,1])
E.analytic_rank(algorithm='pari')

This returns $0$, so the order of the zero at $s=1$ of $L(E,s)$ is $0$ (that is, $L(E,1)$ is nonzero) and hence if BSD holds, we could immediately conclude that $E(\mathbb{Q})$ has rank $0$. Hence assuming BSD, $E$ has only finitely many rational solutions.

Any guidance would be greatly appreciated.

FYI Langlands also discusses icosahedral Galois groups attached to quintic polynomials in the essay but I am not sure if that's relevant.

• If you specifically want to know about rational/integer solutions to Diophantine equations, then I believe Langlands is referring to things like BSD. E.g., look up the congruent number problem. That said, L-functions still tell you other things about Diophantine equations that are hard to realize otherwise, such as Dirichlet's theorem on primes in progressions and what Langlands says about quintic polynomials (e.g., a kind of description of primes $p$ for which a polynomial has a root mod $p$). – Kimball Aug 4 '18 at 1:06
• I had the exact same question when reading that essay! – Teddan the Terran Jun 2 at 16:46