Properties

Degree 3
Conductor $ 2^{4} \cdot 7^{2} $
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 + 2.20·5-s + 0.142·7-s − 0.222·9-s − 11-s − 13-s + 0.733·15-s − 0.764·17-s − 0.789·19-s + 0.0476·21-s + 1.78·23-s + 2.64·25-s + 0.814·27-s − 0.862·29-s − 0.483·31-s − 0.333·33-s + 0.314·35-s − 0.0270·37-s − 0.333·39-s − 0.902·41-s + 0.488·43-s − 0.488·45-s − 0.659·47-s + 0.0204·49-s − 0.254·51-s + 0.886·53-s − 2.20·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 784 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(784\)    =    \(2^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((3,\ 784,\ (1:1.0),\ 1)\)

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-7^{- s})^{-1}\prod_{p \nmid 56 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.764257940\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.435200539\]

Imaginary part of the first few zeros on the critical line

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