Properties

Degree 3
Conductor 289
Sign $1$
Self-dual yes
Motivic weight 2

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Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  − 0.5·2-s − 3-s + 0.750·4-s − 0.200·5-s + 0.5·6-s + 1.285·7-s + 0.375·8-s + 2·9-s + 0.100·10-s − 11-s − 0.750·12-s − 0.692·13-s − 0.642·14-s + 0.200·15-s − 0.312·16-s + 0.058·17-s − 18-s − 0.157·19-s − 0.150·20-s − 1.285·21-s + 0.5·22-s − 0.304·23-s − 0.375·24-s + 0.240·25-s + 0.346·26-s − 2·27-s + 0.964·28-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 289 ^{s/2} \Gamma_{\R}(s+1) \Gamma_{\C}(s+1) \cdot L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{align} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(289\)    =    \(17^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 289,\ (1:1.0),\ 1)$

Euler product

\[\begin{equation} L(s, E, \mathrm{sym}^{2}) = (1-17^{- s})^{-1}\prod_{p \nmid 17 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{equation}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 0.7655251507\] \[L(1, E, \mathrm{sym}^{2}) \approx 0.7850059075\]

Imaginary part of the first few zeros on the critical line

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