Properties

Degree 3
Conductor $ 3^{2} \cdot 19^{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  + 2-s + 0.333·3-s + 0.800·5-s + 0.333·6-s + 2.57·7-s + 0.111·9-s + 0.800·10-s − 0.909·11-s − 0.692·13-s + 2.57·14-s + 0.266·15-s + 16-s − 0.941·17-s + 0.111·18-s + 0.0526·19-s + 0.857·21-s − 0.909·22-s − 0.304·23-s − 0.160·25-s − 0.692·26-s + 0.0370·27-s − 0.862·29-s + 0.266·30-s + 0.161·31-s + 32-s − 0.303·33-s − 0.941·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 3249 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(3249\)    =    \(3^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 3249,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1-19^{- s})^{-1}\prod_{p \nmid 57 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 3.841181158\] \[L(1, E, \mathrm{sym}^{2}) \approx 2.349533923\]

Imaginary part of the first few zeros on the critical line

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