Properties

Degree $3$
Conductor $7056$
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  + 0.333·3-s − 0.200·5-s + 7-s + 0.111·9-s + 2.27·11-s − 0.307·13-s − 0.0666·15-s − 0.0588·17-s + 0.315·19-s + 0.333·21-s − 0.304·23-s + 0.239·25-s + 0.0370·27-s − 0.448·29-s + 0.580·31-s + 0.757·33-s − 0.200·35-s + 1.18·37-s − 0.102·39-s − 0.902·41-s − 0.976·43-s − 0.0222·45-s − 0.914·47-s + 49-s − 0.0196·51-s + 0.207·53-s − 0.454·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 7056 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 7056,\ (1:1.0),\ 1)\)

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.596116394\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.489692307\]

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1+5^{ -s}-5 \ 5^{-2 s}-125 \ 5^{-3 s})^{-1}(1-7\ 7^{- s})^{-1}\prod_{p \nmid 235200 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

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