elliptic curve isogeny class 14.a $L$-function Dirichlet coefficients

Question

• Are the Dirichlet coefficients $a(n)$ of the $L$-function associated with isogeny class 14.a the irrationals that the inverse symbolic calculator suggests they are? The Lcalcfile suggests that they might be:

$$1, -\frac{\sqrt{2}}{2}, -\frac{2\sqrt{3}}{3},\frac{1}{2},0,\sqrt{\frac{2}{3}},\sqrt{\frac{1}{7}},\sqrt{\frac{1}{8}},\frac{1}{3},0,0,\frac{1}{\sqrt{3}},\frac{4}{\sqrt{13}}...$$

• Is there a known closed form for these coefficients?

Answer 1

There are some significant problems with the content of that page. First of all, the $L$-series is usually defined as an Euler product, but the product appearing at the bottom of that page is not correct (when $p$ is good, the Euler factor should be $(1-a(p)p^{-s}+p^{1-2s})^{-1}$. With this definition, then $$L(E,s) = \sum_{n\geq 1} \frac{a(n)}{n^{s}}$$ where we define

• $a(p) = p+1 - \#E(\mathbb{F}_p)$ if $p$ is a prime of good reduction;
• $a(p)= 0$ if $E$ has bad additive reduction at $p$;
• $a(p)=-1$ if $E$ has bad split multiplicative reduction at $p$;
• $a(p)=+1$ if $E$ has bad non-split multiplicative reduction at $p$;
• $a(p^k)a(p)=a(p^{k+1})+p\cdot a(p^{k-1})$ if $E$ has good reduction at $p$;
• $a(p^k)=(a(p))^k$ if $E$ has bad reduction at $p$; and
• $a(nm)=a(n)a(m)$ if $\gcd(n,m)=1$.

For the case of 14a1 you have the following $a(p)$'s: $$ -1, -2, 0, 1, 0, -4, 6, 2, 0, -6, -4, 2, 6, 8, -12, 6, -6, 8, -4, 0, 2, 8, -6, -6, -10, \ldots $$ for $p=2,3,5,7,...$. Thus, the $L$-function should look like: $$L(E,s) = 1 -\frac{1}{2^s}-\frac{2}{3^s}+\frac{1}{4^s}+\frac{2}{6^s}+\cdots$$ However, they seem to be writing the expansion of $$L(E,s+1/2) = 1 - \frac{1}{\sqrt{2}}\cdot \frac{1}{2^s}-\frac{2}{\sqrt{3}}\cdot \frac{1}{3^s}+\frac{1}{2}\cdot \frac{1}{4^s}+\frac{2}{\sqrt{6}}\cdot \frac{1}{6^s}+\cdots$$ But I don't know why. The functional equation is also wrong (there is a factor of $(2\pi)^{-s}$ missing, and it should relate $s$ with $2-s$). All in all, it seems they are trying to write the $L$-series using a different normalization, but they are mixing things up inconsistently in a weird way that is sure to confuse many readers! I'll try to reach out to them and see what is going on.