Properties

Degree $3$
Conductor $43956900$
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 + 0.250·4-s + 0.200·5-s + 0.166·6-s − 0.857·7-s + 0.125·8-s + 0.111·9-s + 0.100·10-s + 1.27·11-s + 0.0833·12-s + 13-s − 0.428·14-s + 0.0666·15-s + 0.0625·16-s + 0.0588·17-s + 0.0555·18-s − 0.947·19-s + 0.0500·20-s − 0.285·21-s + 0.636·22-s − 23-s + 0.0416·24-s + 0.0400·25-s + 0.5·26-s + 0.0370·27-s − 0.214·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 43956900 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(43956900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2} \cdot 17^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 43956900,\ (1:1.0),\ 1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-5^{- s})^{-1}(1-13\ 13^{- s})^{-1}(1-17^{- s})^{-1}\prod_{p \nmid 86190 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

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.