Properties

Degree 4
Conductor $ 3^{4} \cdot 7^{3} \cdot 11^{3} \cdot 17^{4} $
Sign $-1$
Motivic weight 3
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  − 1.06·2-s + 0.375·4-s + 1.07·5-s − 0.0539·7-s − 0.662·8-s − 1.13·10-s − 0.0274·11-s + 1.28·13-s + 0.0572·14-s + 0.546·16-s − 1.06·19-s + 0.402·20-s + 0.0290·22-s + 0.912·25-s − 1.35·26-s − 0.0202·28-s + 0.691·29-s − 0.0926·31-s + 0.580·32-s − 0.0579·35-s + 1.01·37-s + 1.12·38-s − 0.711·40-s + 0.685·41-s + 0.993·43-s − 0.0102·44-s − 0.744·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(3^{4} \cdot 7^{3} \cdot 11^{3} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3^{4} \cdot 7^{3} \cdot 11^{3} \cdot 17^{4}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 3^{4} \cdot 7^{3} \cdot 11^{3} \cdot 17^{4} ,\ ( \ : 1.5, 0.5 ),\ -1 )$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1+7^{ -s})^{-1}(1+11^{ -s})^{-1} \prod_{p \nmid 200277 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.