Properties

Degree $3$
Conductor $1156$
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 − 5-s + 0.166·6-s + 1.28·7-s + 0.125·8-s − 0.222·9-s − 0.5·10-s + 2.27·11-s + 0.0833·12-s − 0.692·13-s + 0.642·14-s − 0.333·15-s + 0.0625·16-s + 0.0588·17-s − 0.111·18-s − 0.157·19-s − 0.250·20-s + 0.428·21-s + 1.13·22-s − 23-s + 0.0416·24-s + 2·25-s − 0.346·26-s + 0.814·27-s + 0.321·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 1156,\ (1:1.0),\ 1)\)

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.027457844\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.548750374\]

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1+5\ 5^{- s}-25 \ 5^{-2 s}-125 \ 5^{-3 s})^{-1}(1-9\ 7^{- s}+63\ 7^{-2 s}-343 \ 7^{-3 s})^{-1}(1-17^{- s})^{-1}\prod_{p \nmid 41650 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

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