Properties

Degree $3$
Conductor $24964$
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  + 0.5·2-s + 2·3-s + 0.250·4-s + 0.800·5-s + 6-s + 0.285·7-s + 0.125·8-s + 2·9-s + 0.400·10-s − 0.636·11-s + 0.5·12-s + 0.923·13-s + 0.142·14-s + 1.60·15-s + 0.0625·16-s + 1.11·17-s + 18-s − 19-s + 0.200·20-s + 0.571·21-s − 0.318·22-s − 0.826·23-s + 0.250·24-s − 0.160·25-s + 0.461·26-s + 27-s + 0.0714·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 24964 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(24964\)    =    \(2^{2} \cdot 79^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 24964,\ (1:1.0),\ 1)\)

Particular Values

L(1/2): not computed L(1): not computed

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-79^{- s})^{-1}\prod_{p \nmid 158 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

Zeros not available.

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

Plot not available.