Properties

Degree 4
Conductor $ 2^{14} $
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  + 0.769·3-s + 1.07·5-s − 0.431·7-s + 0.814·9-s − 0.986·11-s + 0.938·13-s + 0.826·15-s + 0.856·17-s + 0.821·19-s − 0.332·21-s − 1.08·23-s + 0.912·25-s + 1.56·27-s − 0.845·29-s − 0.759·33-s − 0.463·35-s − 1.15·37-s + 0.722·39-s + 1.05·41-s + 1.06·43-s + 0.874·45-s + 0.744·47-s − 0.180·49-s + 0.658·51-s − 1.08·53-s − 1.05·55-s + 0.632·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(16384\)    =    \(2^{14}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 16384,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = \prod_{p \nmid 128 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

L(1/2): 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.