Properties

Degree 4
Conductor $ 11^{4} $
Sign $-1$
Motivic weight 3
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  + 0.962·3-s − 0.804·5-s − 1.07·7-s − 0.185·9-s + 0.853·13-s − 0.774·15-s − 16-s − 0.856·17-s − 1.03·21-s + 0.407·23-s − 0.792·25-s − 0.285·27-s − 0.527·31-s + 0.869·35-s − 0.866·37-s + 0.821·39-s − 0.548·41-s − 1.06·43-s + 0.149·45-s − 0.744·47-s − 0.962·48-s + 0.553·49-s − 0.823·51-s + 1.08·53-s − 1.02·59-s − 0.554·61-s + 0.199·63-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(14641\)    =    \(11^{4}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 14641,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = \prod_{p \nmid 121 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 0\] \[L(1, E, \mathrm{sym}^{3}) \approx 0.7926318373\]

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line