Properties

Degree 3
Conductor 1369
Sign $1$
Self-dual yes
Motivic weight 2

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Normalization:  

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Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  − 2-s − 0.666·3-s + 2·4-s − 5-s + 0.666·6-s − 0.857·7-s − 2·8-s + 1.111·9-s + 10-s − 0.181·11-s − 1.333·12-s + 0.230·13-s + 0.857·14-s + 0.666·15-s + 3·16-s + 1.117·17-s − 1.111·18-s − 0.789·19-s − 2·20-s + 0.571·21-s + 0.181·22-s + 0.565·23-s + 1.333·24-s + 2·25-s − 0.230·26-s − 0.185·27-s − 1.714·28-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 1369 ^{s/2} \Gamma_{\R}(s+1) \Gamma_{\C}(s+1) \cdot L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{align} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(1369\)    =    \(37^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 1369,\ (1:1.0),\ 1)$

Euler product

\[\begin{equation} L(s, E, \mathrm{sym}^{2}) = (1-37^{- s})^{-1}\prod_{p \nmid 37 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{equation}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 0.7728058998\] \[L(1, E, \mathrm{sym}^{2}) \approx 0.6534792516\]

Imaginary part of the first few zeros on the critical line

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