Properties

Degree 3
Conductor $ 11^{2} \cdot 13^{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 − 0.666·3-s + 2·4-s − 0.800·5-s + 0.666·6-s − 0.428·7-s − 2·8-s + 1.11·9-s + 0.800·10-s + 0.0909·11-s − 1.33·12-s + 0.0769·13-s + 0.428·14-s + 0.533·15-s + 3·16-s − 0.0588·17-s − 1.11·18-s − 0.789·19-s − 1.60·20-s + 0.285·21-s − 0.0909·22-s + 1.13·23-s + 1.33·24-s + 1.43·25-s − 0.0769·26-s − 0.185·27-s − 0.857·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 20449 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(20449\)    =    \(11^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 20449,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-11^{- s})^{-1}(1-13^{- s})^{-1}\prod_{p \nmid 143 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.