Properties

Degree $3$
Conductor $441$
Sign $1$
Motivic weight $2$
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  + 2-s + 0.333·3-s − 0.200·5-s + 0.333·6-s + 7-s + 0.111·9-s − 0.200·10-s − 0.636·11-s − 0.923·13-s + 14-s − 0.0666·15-s + 16-s − 17-s + 0.111·18-s − 0.947·19-s + 0.333·21-s − 0.636·22-s − 23-s + 0.239·25-s − 0.923·26-s + 0.0370·27-s − 0.448·29-s − 0.0666·30-s + 1.61·31-s + 32-s − 0.212·33-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 441,\ (1:1.0),\ 1)\)

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.797737269\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.621788676\]

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-2\ 2^{- s}+4\ 2^{-2 s}-8 \ 2^{-3 s})^{-1}(1-3^{- s})^{-1}(1-7\ 7^{- s})^{-1}\prod_{p \nmid 9408 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

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