Properties

Degree 3
Conductor $ 2^{2} \cdot 71^{2} $
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 − 0.666·3-s + 0.250·4-s − 5-s − 0.333·6-s − 0.857·7-s + 0.125·8-s + 1.11·9-s − 0.5·10-s − 11-s − 0.166·12-s − 0.923·13-s − 0.428·14-s + 0.666·15-s + 0.0625·16-s − 17-s + 0.555·18-s − 0.947·19-s − 0.250·20-s + 0.571·21-s − 0.5·22-s − 0.608·23-s − 0.0833·24-s + 2·25-s − 0.461·26-s − 0.185·27-s − 0.214·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 20164 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(20164\)    =    \(2^{2} \cdot 71^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 20164,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-71^{- s})^{-1}\prod_{p \nmid 142 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

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

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.