Properties

Degree $6$
Conductor $2476099$
Sign $-1$
Motivic weight $5$
Arithmetic yes
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{5})$  = 1  + 0.641·3-s − 3·4-s − 1.28·5-s − 0.925·7-s − 0.267·9-s + 0.358·11-s − 1.92·12-s + 0.452·13-s − 0.826·15-s + 6·16-s − 0.845·17-s + 0.000635·19-s + 3.86·20-s − 0.593·21-s + 0.545·25-s + 0.960·27-s + 2.77·28-s − 0.472·29-s − 0.863·31-s + 0.230·33-s + 1.19·35-s + 0.802·36-s + 0.848·37-s + 0.290·39-s − 0.242·41-s − 0.443·43-s − 1.07·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{5})=\mathstrut & 2476099 ^{s/2} \, \Gamma_{\C}(s+2.5) \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{5})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{5}) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2476099\)    =    \(19^{5}\)
Sign: $-1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((6,\ 2476099,\ (\ :2.5, 1.5, 0.5),\ -1)\)

Particular Values

L(1/2): not computed L(1): not computed

Euler product

\(L(s, E, \mathrm{sym}^{5}) = (1-19^{- s})^{-1}\prod_{p \nmid 19 }\prod_{j=0}^{5} \left(1- \frac{\alpha_p^j\beta_p^{5-j}}{p^{s}} \right)^{-1}\)

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.