Properties

Degree $3$
Conductor $1764$
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 + 0.142·7-s + 0.125·8-s + 0.111·9-s − 0.100·10-s + 0.454·11-s + 0.0833·12-s + 1.76·13-s + 0.0714·14-s − 0.0666·15-s + 0.0625·16-s − 0.764·17-s + 0.0555·18-s − 0.157·19-s − 0.0500·20-s + 0.0476·21-s + 0.227·22-s + 1.78·23-s + 0.0416·24-s + 0.239·25-s + 0.884·26-s + 0.0370·27-s + 0.0357·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 1764,\ (1:1.0),\ 1)\)

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.315608640\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.639831976\]

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 42 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

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