Properties

Degree 5
Conductor $ 2^{6} \cdot 7^{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  + 3-s − 0.760·5-s + 0.0204·7-s + 3·9-s − 1.24·11-s + 0.171·13-s − 0.760·15-s − 0.868·17-s + 2.24·19-s + 0.0204·21-s + 23-s − 0.334·25-s + 3·27-s − 1.18·29-s − 0.931·31-s − 1.24·33-s − 0.0155·35-s + 0.687·37-s + 0.171·39-s + 0.716·41-s + 0.0221·43-s − 2.27·45-s − 1.23·47-s + 0.000416·49-s − 0.868·51-s − 0.576·53-s + 0.948·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 153664 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned} \]

Invariants

\( d \)  =  \(5\)
\( N \)  =  \(153664\)    =    \(2^{6} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(5,\ 153664,\ (0:2.0, 1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{4}) = (1-7^{- s})^{-1}\prod_{p \nmid 112 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.