Properties

Degree 6
Conductor $ 2^{12} \cdot 3^{5} \cdot 5^{6} \cdot 11^{6} $
Sign $-1$
Motivic weight 5
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{5})$  = 1  + 0.0641·3-s − 0.786·7-s + 0.00411·9-s + 0.452·13-s + 1.43·17-s − 0.312·19-s − 0.0504·21-s − 1.01·23-s + 0.000263·27-s + 0.472·29-s + 1.43·31-s − 0.832·37-s + 0.0290·39-s − 0.242·41-s + 0.900·43-s − 0.457·47-s + 1.02·49-s + 0.0920·51-s − 0.0200·57-s + 0.0982·61-s − 0.00323·63-s − 1.02·67-s − 0.0650·69-s − 0.875·71-s + 0.651·73-s − 1.21·79-s + 1.69e−5·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{5})=\mathstrut &\left(2^{12} \cdot 3^{5} \cdot 5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+2.5) \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{5})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{5})\end{aligned}\]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(2^{12} \cdot 3^{5} \cdot 5^{6} \cdot 11^{6}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((6,\ 2^{12} \cdot 3^{5} \cdot 5^{6} \cdot 11^{6} ,\ ( \ : 2.5, 1.5, 0.5 ),\ -1 )\)

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{5}) = (1-3^{- s})^{-1} \prod_{p \nmid 145200 }\prod_{j=0}^{5} \left(1- \frac{\alpha_p^j\beta_p^{5-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.