Properties

Degree 3
Conductor $ 53^{2} $
Sign $1$
Motivic weight 2
Primitive yes
Self-dual yes

Related objects

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Normalization:  

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Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  − 0.5·2-s + 2·3-s + 0.750·4-s − 5-s − 6-s + 1.28·7-s + 0.375·8-s + 2·9-s + 0.5·10-s − 11-s + 1.5·12-s − 0.307·13-s − 0.642·14-s − 2·15-s − 0.312·16-s − 0.470·17-s − 18-s + 0.315·19-s − 0.750·20-s + 2.57·21-s + 0.5·22-s + 1.13·23-s + 0.750·24-s + 2·25-s + 0.153·26-s + 27-s + 0.964·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 2809 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(2809\)    =    \(53^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 2809,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-53^{- s})^{-1}\prod_{p \nmid 53 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.672168913\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.712283242\]

Imaginary part of the first few zeros on the critical line

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