Properties

Degree 5
Conductor $ 2^{4} \cdot 3^{4} \cdot 11^{4} $
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 + 5-s + 0.0277·6-s − 0.387·7-s + 0.0156·8-s + 0.0123·9-s + 0.250·10-s + 0.00826·11-s + 0.00694·12-s − 1.17·13-s − 0.0969·14-s + 0.111·15-s + 0.00390·16-s − 0.868·17-s + 0.00308·18-s − 0.817·19-s + 0.0625·20-s − 0.0430·21-s + 0.00206·22-s − 1.24·23-s + 0.00173·24-s + 3·25-s − 0.294·26-s + 0.00137·27-s − 0.0242·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 18974736 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned}\]

Invariants

\( d \)  =  \(5\)
\( N \)  =  \(18974736\)    =    \(2^{4} \cdot 3^{4} \cdot 11^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((5,\ 18974736,\ (0:2.0, 1.0),\ 1)\)

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{4}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-25\ 5^{- s}-1250 \ 5^{-2 s}+31250\ 5^{-3 s}+390625\ 5^{-4 s}-9765625 \ 5^{-5 s})^{-1}(1-11^{- s})^{-1}\prod_{p \nmid 145200 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.