# Properties

 Degree $5$ Conductor $3.833\times 10^{17}$ Sign $1$ Motivic weight $4$ Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{4})$  = 1 + 0.250·2-s + 0.111·3-s + 0.0625·4-s + 0.440·5-s + 0.0277·6-s + 0.591·7-s + 0.0156·8-s + 0.0123·9-s + 0.110·10-s + 0.00826·11-s + 0.00694·12-s + 0.00591·13-s + 0.147·14-s + 0.0488·15-s + 0.00390·16-s − 0.307·17-s + 0.00308·18-s + 0.844·19-s + 0.0275·20-s + 0.0657·21-s + 0.00206·22-s − 0.0207·23-s + 0.00173·24-s + 0.985·25-s + 0.00147·26-s + 0.00137·27-s + 0.0369·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4} \cdot 29^{4}\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: $$2^{4} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4} \cdot 29^{4}$$ Sign: $1$ Primitive: yes Self-dual: yes Selberg data: $$(5,\ 2^{4} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4} \cdot 29^{4} ,\ ( 0 : 2.0, 1.0 ),\ 1 )$$

## Particular Values

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

## Euler product

$$L(s, E, \mathrm{sym}^{4}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-11^{- s})^{-1}(1-13^{- s})^{-1}(1-29^{- s})^{-1}\prod_{p \nmid 24882 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}$$

## 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.