Properties

Degree $5$
Conductor $5184$
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  + 0.111·3-s − 0.760·5-s + 7-s + 0.0123·9-s − 1.24·11-s + 0.171·13-s − 0.0844·15-s + 0.349·17-s − 0.817·19-s + 0.111·21-s + 0.395·23-s − 0.334·25-s + 0.00137·27-s − 1.18·29-s − 0.931·31-s − 0.138·33-s − 0.760·35-s − 0.972·37-s + 0.0190·39-s − 0.863·41-s + 0.0221·43-s − 0.00938·45-s + 47-s + 3·49-s + 0.0388·51-s + 0.779·53-s + 0.948·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned}\]

Invariants

Degree: \(5\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((5,\ 5184,\ (0:2.0, 1.0),\ 1)\)

Particular Values

\[L(1/2, E, \mathrm{sym}^{4}) \approx 0.6146164707\] \[L(1, E, \mathrm{sym}^{4}) \approx 0.8771625213\]

Euler product

\(L(s, E, \mathrm{sym}^{4}) = (1-3^{- s})^{-1}\prod_{p \nmid 48 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line