Properties

Degree $3$
Conductor $44100$
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 − 3-s + 0.250·4-s + 0.200·5-s − 0.5·6-s − 7-s + 0.125·8-s + 9-s + 0.100·10-s − 0.636·11-s − 0.250·12-s − 0.692·13-s − 0.5·14-s − 0.200·15-s + 0.0625·16-s − 0.764·17-s + 0.5·18-s + 0.894·19-s + 0.0500·20-s + 21-s − 0.318·22-s − 0.304·23-s − 0.125·24-s + 0.0400·25-s − 0.346·26-s − 27-s − 0.250·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 44100 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 44100,\ (1:1.0),\ 1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1+3\ 3^{- s})^{-1}(1-5^{- s})^{-1}(1+7\ 7^{- s})^{-1}\prod_{p \nmid 4410 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

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.