Properties

Degree $3$
Conductor $6084$
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 + 0.333·3-s + 4-s − 5-s − 0.333·6-s − 0.428·7-s − 8-s + 0.111·9-s + 10-s − 11-s + 0.333·12-s + 0.0769·13-s + 0.428·14-s − 0.333·15-s + 16-s + 1.11·17-s − 0.111·18-s − 0.789·19-s − 20-s − 0.142·21-s + 22-s − 23-s − 0.333·24-s + 2·25-s − 0.0769·26-s + 0.0370·27-s − 0.428·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 6084 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(6084\)    =    \(2^{2} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 6084,\ (1:1.0),\ 1)\)

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 0.7572500090\] \[L(1, E, \mathrm{sym}^{2}) \approx 0.6067157552\]

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1+2\ 2^{- s})^{-1}(1-3^{- s})^{-1}(1-13^{- s})^{-1}\prod_{p \nmid 156 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

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