Properties

Degree 5
Conductor $ 3^{4} \cdot 19^{4} $
Sign $1$
Motivic weight 4
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{4})$  = 1  − 2-s + 0.111·3-s + 4-s − 1.15·5-s − 0.111·6-s + 3.04·7-s − 8-s + 0.0123·9-s + 1.15·10-s + 0.735·11-s + 0.111·12-s + 0.171·13-s − 3.04·14-s − 0.128·15-s + 2·16-s + 0.826·17-s − 0.0123·18-s + 0.00277·19-s − 1.15·20-s + 0.337·21-s − 0.735·22-s − 0.603·23-s − 0.111·24-s + 1.11·25-s − 0.171·26-s + 0.00137·27-s + 3.04·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 10556001 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned} \]

Invariants

\( d \)  =  \(5\)
\( N \)  =  \(10556001\)    =    \(3^{4} \cdot 19^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(5,\ 10556001,\ (0:2.0, 1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{4}) = (1-3^{- s})^{-1}(1-19^{- s})^{-1}\prod_{p \nmid 57 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.