Properties

Degree 3
Conductor $ 179^{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 − 3-s + 0.800·5-s − 6-s + 1.28·7-s + 2·9-s + 0.800·10-s + 0.454·11-s − 0.923·13-s + 1.28·14-s − 0.800·15-s + 16-s − 0.941·17-s + 2·18-s − 0.526·19-s − 1.28·21-s + 0.454·22-s + 0.565·23-s − 0.160·25-s − 0.923·26-s − 2·27-s − 0.689·29-s − 0.800·30-s + 1.06·31-s + 32-s − 0.454·33-s − 0.941·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 32041 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(32041\)    =    \(179^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 32041,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-179^{- s})^{-1}\prod_{p \nmid 179 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.