Properties

Degree $3$
Conductor $1225$
Sign $1$
Motivic weight $2$
Arithmetic yes
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  + 2-s + 2·3-s + 0.200·5-s + 2·6-s − 7-s + 2·9-s + 0.200·10-s − 0.909·11-s − 0.307·13-s − 14-s + 0.400·15-s + 16-s − 0.470·17-s + 2·18-s + 0.894·19-s − 2·21-s − 0.909·22-s − 0.304·23-s + 0.0400·25-s − 0.307·26-s + 27-s − 0.965·29-s + 0.400·30-s + 0.161·31-s + 32-s − 1.81·33-s − 0.470·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 1225,\ (1:1.0),\ 1)\)

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 3.424883075\] \[L(1, E, \mathrm{sym}^{2}) \approx 2.478888106\]

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-5^{- s})^{-1}(1+7\ 7^{- s})^{-1}\prod_{p \nmid 245 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

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