Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We have a $L$-function $$L(E,s)$$ built from the local parameters $a_p(E)$. If two elliptic curves are isomorphic, they clearly have the same $L$-function. What about the converse ? If two elliptic curves $E,E’$ over $\mathbb{Q}$ have the same $L$-function, what can be said about them ? […]

Let $f:\mathbb{N}\to\mathbb{C}$ and assume that the L-function $$L(f;s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}$$ converges absolutely on some right half-plane $Re(s)>k$. Is it true that $L(f;\cdot)$ is identically zero if and only if so is $f$? If not, is there any classification of such arithmetic functions? My guess is that there exists a non-zero function $f$ with zero L-function, but […]

I haven’t been able to find a reference that tells what word (if a word) the L is short for.

Consider the two Dirichlet characters of $\mathbb{Z}/3\mathbb{Z}$. $$ \begin{array}{c|ccr} & 0 & 1 & 2 \\ \hline \chi_1 & 0 & 1 & 1 \\ \chi_2 & 0 & 1 & -1 \end{array} $$ I read the L-functions for these series have special values $ L(2,\chi_1) \in \pi^2 \sqrt{3}\;\mathbb{Q} $ $ L(1,\chi_2) \in \pi \sqrt{3}\;\mathbb{Q} […]

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