Properties

Degree 2
Level 389
Sign $1$
Self-dual
Motivic weight 1

Related objects

Downloads

Dirichlet series

$L(s,E) = 1^{\mathstrut}$  − 1.414 2-s − 1.154 3-s + 4-s − 1.341 5-s + 1.632 6-s − 1.889 7-s + 0.333 9-s + 1.897 10-s − 1.206 11-s − 1.154 12-s − 0.832 13-s + 2.672 14-s + 1.549 15-s − 16-s − 1.455 17-s − 0.471 18-s + 1.147 19-s − 1.341 20-s + 2.182 21-s + 1.705 22-s − 0.834 23-s + 0.8 25-s + 1.176 26-s + 0.769 27-s − 1.889 28-s − 1.114 29-s − 2.190 30-s + ...

Functional equation

\[\begin{align} \Lambda(s,E)=\mathstrut & 389 ^{s/2} \Gamma_{\C}(s+1/2) \cdot L(s,E)\cr =\mathstrut & \Lambda(1-s,E) \end{align} \]
Selberg data: $(2,389,(:1/2), 1)$

Euler product

\[\begin{equation} L(s,E) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + p^{-2s})^{-1} \end{equation}\]

Imaginary part of the first few zeros on the critical line

Particular Values

\[L(1/2,E)=0\]
\[L(1,E) \approx 0.1337684325\]

Graph of the $Z$-function along center part of the critical line