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  + 1.06·2-s − 0.769·3-s + 0.375·4-s − 0.804·5-s − 0.816·6-s − 1.07·7-s + 0.662·8-s + 0.814·9-s − 0.853·10-s − 0.288·12-s + 0.533·13-s − 1.14·14-s + 0.619·15-s + 0.546·16-s − 0.642·17-s + 0.864·18-s + 0.144·19-s − 0.301·20-s + 0.831·21-s − 0.761·23-s − 0.510·24-s − 0.792·25-s + 0.565·26-s − 1.56·27-s − 0.404·28-s − 1.32·29-s + 0.657·30-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

\[L(1/2, E, \mathrm{sym}^{3}) \approx 0\] \[L(1, E, \mathrm{sym}^{3}) \approx 0.8558322879\]

Imaginary part of the first few zeros on the critical line

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