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 + 0.800·5-s − 0.5·6-s + 1.28·7-s + 0.125·8-s + 9-s + 0.400·10-s − 11-s − 0.250·12-s − 0.923·13-s + 0.642·14-s − 0.800·15-s + 0.0625·16-s − 0.470·17-s + 0.5·18-s − 0.157·19-s + 0.200·20-s − 1.28·21-s − 0.5·22-s − 23-s − 0.125·24-s − 0.160·25-s − 0.461·26-s − 27-s + 0.321·28-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

\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.181496032\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.153866273\]

Imaginary part of the first few zeros on the critical line

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