Properties

Degree 6
Conductor $ 37^{5} $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{5})$  = 1  + 1.41·2-s + 4-s − 0.393·5-s − 0.925·7-s − 0.556·10-s + 1.39·11-s − 1.03·13-s − 1.30·14-s − 2·16-s − 0.393·20-s + 1.97·22-s + 0.973·23-s + 0.489·25-s − 1.46·26-s − 0.925·28-s − 0.472·29-s − 0.863·31-s − 2.82·32-s + 0.364·35-s − 0.000120·37-s + 1.40·41-s + 0.804·43-s + 1.39·44-s + 1.37·46-s + 1.21·47-s − 0.592·49-s + 0.691·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{5})=\mathstrut & 69343957 ^{s/2} \, \Gamma_{\C}(s+2.5) \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{5})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{5}) \end{aligned}\]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(69343957\)    =    \(37^{5}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((6,\ 69343957,\ (\ :2.5, 1.5, 0.5),\ 1)\)

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{5}) = (1+37^{ -s})^{-1}\prod_{p \nmid 37 }\prod_{j=0}^{5} \left(1- \frac{\alpha_p^j\beta_p^{5-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.