Properties

Degree 4
Conductor $ 3^{4} \cdot 5^{3} \cdot 7^{3} $
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 − 0.0894·5-s + 0.0539·7-s + 0.662·8-s − 0.0948·10-s − 1.28·13-s + 0.0572·14-s + 0.546·16-s + 0.856·17-s − 2.51·19-s − 0.0335·20-s − 1.30·23-s + 0.00800·25-s − 1.35·26-s + 0.0202·28-s − 0.691·29-s − 1.06·31-s − 0.580·32-s + 0.907·34-s − 0.00482·35-s + 0.622·37-s − 2.66·38-s − 0.0592·40-s − 1.05·41-s − 0.993·43-s − 1.38·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 3472875 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3472875\)    =    \(3^{4} \cdot 5^{3} \cdot 7^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 3472875,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1+5^{ -s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 315 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.