Properties

Degree $5$
Conductor $3111696$
Sign $1$
Motivic weight $4$
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{4})$  = 1  + 0.250·2-s + 0.111·3-s + 0.0625·4-s − 0.760·5-s + 0.0277·6-s + 0.0204·7-s + 0.0156·8-s + 0.0123·9-s − 0.190·10-s − 1.24·11-s + 0.00694·12-s + 0.360·13-s + 0.00510·14-s − 0.0844·15-s + 0.00390·16-s + 0.349·17-s + 0.00308·18-s − 0.817·19-s − 0.0475·20-s + 0.00226·21-s − 0.311·22-s + 0.395·23-s + 0.00173·24-s − 0.334·25-s + 0.0902·26-s + 0.00137·27-s + 0.00127·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned}\]

Invariants

Degree: \(5\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((5,\ 3111696,\ (0:2.0, 1.0),\ 1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{4}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 42 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

Zeros not available.

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

Plot not available.