Properties

Degree $3$
Conductor $1296$
Sign $1$
Motivic weight $2$
Arithmetic yes
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  + 2.20·5-s + 0.285·7-s + 0.454·11-s − 0.923·13-s − 0.0588·17-s − 0.947·19-s − 0.304·23-s + 2.64·25-s − 29-s − 0.483·31-s + 0.628·35-s + 1.18·37-s − 41-s + 0.488·43-s + 2.06·47-s − 0.204·49-s + 0.207·53-s + 55-s − 0.728·59-s − 0.590·61-s − 2.03·65-s + 0.805·67-s − 0.0985·71-s − 0.986·73-s + 0.129·77-s − 0.683·79-s − 0.228·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 1296 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 1296,\ (1:1.0),\ 1)\)

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.956234985\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.456219285\]

Euler product

\(L(s, E, \mathrm{sym}^{2}) = \prod_{p \nmid 216 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

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