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 + 2·3-s − 0.200·5-s + 2·6-s − 0.857·7-s + 2·9-s − 0.200·10-s + 1.272·11-s − 0.692·13-s − 0.857·14-s − 0.400·15-s + 16-s − 17-s + 2·18-s − 19-s − 1.714·21-s + 1.272·22-s − 0.826·23-s + 0.240·25-s − 0.692·26-s + 27-s + 0.241·29-s − 0.400·30-s − 0.483·31-s + 32-s + 2.545·33-s − 34-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

\[L(1/2, E, \mathrm{sym}^{2}) \approx 3.5133942238\] \[L(1, E, \mathrm{sym}^{2}) \approx 2.4922620443\]

Imaginary part of the first few zeros on the critical line

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