Properties

Degree 4
Conductor 1331
Sign $1$
Self-dual yes
Motivic weight 3

Related objects

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Normalization:  

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Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  + 0.962·3-s − 0.804·5-s + 1.079·7-s − 0.185·9-s + 0.027·11-s − 0.853·13-s − 0.774·15-s − 16-s + 0.856·17-s + 1.039·21-s + 0.407·23-s − 0.792·25-s − 0.285·27-s − 0.527·31-s + 0.026·33-s − 0.869·35-s − 0.866·37-s − 0.821·39-s + 0.548·41-s + 1.063·43-s + 0.149·45-s − 0.744·47-s − 0.962·48-s + 0.553·49-s + 0.823·51-s + 1.088·53-s − 0.022·55-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 1331 ^{s/2} \Gamma_{\C}(s+1.5) \Gamma_{\C}(s+0.5) \cdot L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{align} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1331\)    =    \(11^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 1331,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{equation} L(s, E, \mathrm{sym}^{3}) = (1-11^{- s})^{-1}\prod_{p \nmid 11 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{equation}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 1.1402308684\] \[L(1, E, \mathrm{sym}^{3}) \approx 1.1763280759\]

Imaginary part of the first few zeros on the critical line

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