Properties

Degree 5
Conductor $ 3^{4} \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  − 0.250·2-s + 0.111·3-s − 0.312·4-s − 0.760·5-s − 0.0277·6-s + 0.0204·7-s + 0.546·8-s + 0.0123·9-s + 0.190·10-s − 1.24·11-s − 0.0347·12-s + 0.171·13-s − 0.00510·14-s − 0.0844·15-s + 0.136·16-s − 0.868·17-s − 0.00308·18-s − 0.817·19-s + 0.237·20-s + 0.00226·21-s + 0.311·22-s + 23-s + 0.0607·24-s − 0.334·25-s − 0.0428·26-s + 0.00137·27-s − 0.00637·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 194481 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned}\]

Invariants

\( d \)  =  \(5\)
\( N \)  =  \(194481\)    =    \(3^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(5,\ 194481,\ (0:2.0, 1.0),\ 1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{4}) = (1-3^{- s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 21 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.