# Properties

 Degree 5 Conductor $2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}$ Sign $1$ Motivic weight 4 Primitive yes Self-dual yes

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## 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 + 0.0204·7-s + 0.0156·8-s + 0.0123·9-s + 0.0100·10-s + 0.00826·11-s + 0.00694·12-s + 0.171·13-s + 0.00510·14-s + 0.00444·15-s + 0.00390·16-s − 0.868·17-s + 0.00308·18-s − 0.817·19-s + 0.00250·20-s + 0.00226·21-s + 0.00206·22-s + 23-s + 0.00173·24-s + 0.00160·25-s + 0.0428·26-s + 0.00137·27-s + 0.00127·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{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

 $$d$$ = $$5$$ $$N$$ = $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(5,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( 0 : 2.0, 1.0 ),\ 1 )$

## Euler product

\begin{aligned}L(s, E, \mathrm{sym}^{4}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-5^{- s})^{-1}(1-7^{- s})^{-1}(1-11^{- s})^{-1}\prod_{p \nmid 2310 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\end{aligned}

## Particular Values

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

## 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.