Properties

Degree 3
Conductor $ 3^{2} \cdot 11^{2} $
Sign $1$
Motivic weight 2
Primitive yes
Self-dual yes

Related objects

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Normalization:  

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Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  + 2-s + 0.333·3-s + 2.20·5-s + 0.333·6-s − 0.857·7-s + 0.111·9-s + 2.20·10-s − 11-s − 0.692·13-s − 0.857·14-s + 0.733·15-s + 16-s − 0.0588·17-s + 0.111·18-s − 0.526·19-s − 0.285·21-s − 22-s − 0.826·23-s + 2.64·25-s − 0.692·26-s + 0.0370·27-s + 0.241·29-s + 0.733·30-s − 0.193·31-s + 32-s − 0.333·33-s − 0.0588·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(1089\)    =    \(3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((3,\ 1089,\ (1:1.0),\ 1)\)

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1+11\ 11^{- s})^{-1}(1+9\ 13^{- s}-117 \ 13^{-2 s}-2197 \ 13^{-3 s})^{-1}\prod_{p \nmid 61347 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.717100924\] \[L(1, E, \mathrm{sym}^{2}) \approx 2.039699902\]

Imaginary part of the first few zeros on the critical line

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