Properties

Degree $5$
Conductor $292410000$
Sign $1$
Motivic weight $4$
Arithmetic yes
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 + 0.0400·5-s + 0.0277·6-s − 0.632·7-s + 0.0156·8-s + 0.0123·9-s + 0.0100·10-s + 1.89·11-s + 0.00694·12-s − 1.17·13-s − 0.158·14-s + 0.00444·15-s + 0.00390·16-s − 0.868·17-s + 0.00308·18-s + 19-s + 0.00250·20-s − 0.0702·21-s + 0.473·22-s − 1.24·23-s + 0.00173·24-s + 0.00160·25-s − 0.294·26-s + 0.00137·27-s − 0.0395·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{2}\right)^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{4})\end{aligned}\]

Invariants

Degree: \(5\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((5,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{2} ,\ ( 0 : 2.0, 1.0 ),\ 1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{4}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-5^{- s})^{-1}(1-361\ 19^{- s}-130321 \ 19^{-2 s}+47045881\ 19^{-3 s})^{-1}\prod_{p \nmid 10830 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\)

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.