Properties

Degree $3$
Conductor $4900$
Sign $1$
Motivic weight $2$
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  − 2-s + 2·3-s + 4-s + 0.200·5-s − 2·6-s + 0.142·7-s − 8-s + 2·9-s − 0.200·10-s + 1.27·11-s + 2·12-s − 0.307·13-s − 0.142·14-s + 0.400·15-s + 16-s − 0.941·17-s − 2·18-s + 0.894·19-s + 0.200·20-s + 0.285·21-s − 1.27·22-s + 0.565·23-s − 2·24-s + 0.0400·25-s + 0.307·26-s + 27-s + 0.142·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 4900 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 4900,\ (1:1.0),\ 1)\)

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.539693795\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.502676144\]

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1+2\ 2^{- s})^{-1}(1-6\ 3^{- s}+18\ 3^{-2 s}-27 \ 3^{-3 s})^{-1}(1-5^{- s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 25200 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

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