Properties

Degree 5
Conductor $ 37^{4} $
Sign $1$
Motivic weight 4
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{4})$  = 1  − 2-s + 3-s + 4-s − 0.760·5-s − 6-s + 0.591·7-s − 8-s + 0.760·10-s − 0.652·11-s + 12-s + 0.171·13-s − 0.591·14-s − 0.760·15-s + 2·16-s + 17-s + 19-s − 0.760·20-s + 0.591·21-s + 0.652·22-s + 0.508·23-s − 24-s − 0.334·25-s − 0.171·26-s + 0.591·28-s − 1.18·29-s + 0.760·30-s − 0.281·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 1874161 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned} \]

Invariants

\( d \)  =  \(5\)
\( N \)  =  \(1874161\)    =    \(37^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(5,\ 1874161,\ (0:2.0, 1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{4}) = (1-37^{- s})^{-1}\prod_{p \nmid 37 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.