Properties

Degree $3$
Conductor $439569$
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 + 0.333·3-s + 0.750·4-s − 0.200·5-s − 0.166·6-s − 7-s + 0.375·8-s + 0.111·9-s + 0.100·10-s + 0.454·11-s + 0.250·12-s + 0.0769·13-s + 0.5·14-s − 0.0666·15-s − 0.312·16-s + 0.0588·17-s − 0.0555·18-s − 0.157·19-s − 0.149·20-s − 0.333·21-s − 0.227·22-s − 23-s + 0.125·24-s + 0.239·25-s − 0.0384·26-s + 0.0370·27-s − 0.750·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 439569 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(439569\)    =    \(3^{2} \cdot 13^{2} \cdot 17^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 439569,\ (1:1.0),\ 1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1-13^{- s})^{-1}(1-17^{- s})^{-1}\prod_{p \nmid 663 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

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.