Properties

Degree 4
Conductor $ 2^{3} \cdot 3^{5} $
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  + 0.353·2-s + 0.125·4-s + 0.268·5-s + 0.701·7-s + 0.0441·8-s + 0.0948·10-s − 1.06·11-s + 0.853·13-s + 0.248·14-s + 0.0156·16-s − 0.821·19-s + 0.0335·20-s − 0.377·22-s − 0.543·23-s + 0.232·25-s + 0.301·26-s + 0.0877·28-s + 0.845·29-s − 1.07·31-s + 0.00552·32-s + 0.188·35-s − 0.622·37-s − 0.290·38-s + 0.0118·40-s − 1.05·41-s − 0.496·43-s − 0.133·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1944\)    =    \(2^{3} \cdot 3^{5}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 1944,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1} \prod_{p \nmid 54 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 1.324322392\] \[L(1, E, \mathrm{sym}^{3}) \approx 1.271130391\]

Imaginary part of the first few zeros on the critical line

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