Properties

Degree 3
Conductor $ 2^{2} \cdot 3^{4} $
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 + 3-s + 0.250·4-s − 5-s + 0.5·6-s − 0.428·7-s + 0.125·8-s + 9-s − 0.5·10-s − 0.181·11-s + 0.250·12-s − 0.692·13-s − 0.214·14-s − 15-s + 0.0625·16-s − 0.470·17-s + 0.5·18-s − 0.947·19-s − 0.250·20-s − 0.428·21-s − 0.0909·22-s + 0.565·23-s + 0.125·24-s + 2·25-s − 0.346·26-s + 27-s − 0.107·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 324,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-3\ 3^{- s})^{-1}\prod_{p \nmid 162 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.483268379\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.425723351\]

Imaginary part of the first few zeros on the critical line

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