Properties

Degree 3
Conductor $ 2^{2} \cdot 7537^{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 + 2·3-s + 4-s + 2.20·5-s − 2·6-s + 1.28·7-s − 8-s + 2·9-s − 2.20·10-s − 0.636·11-s + 2·12-s − 0.692·13-s − 1.28·14-s + 4.40·15-s + 16-s + 2.76·17-s − 2·18-s + 0.894·19-s + 2.20·20-s + 2.57·21-s + 0.636·22-s + 0.0869·23-s − 2·24-s + 2.64·25-s + 0.692·26-s + 27-s + 1.28·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut &\left(2^{2} \cdot 7537^{2}\right)^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{2})\end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(2^{2} \cdot 7537^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 2^{2} \cdot 7537^{2} ,\ ( 1 : 1.0 ),\ 1 )$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1+2\ 2^{- s})^{-1}(1-7537^{- s})^{-1}\prod_{p \nmid 30148 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.