Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{3} $
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  − 0.192·3-s + 1.07·5-s + 0.0370·9-s − 0.657·11-s + 0.938·13-s − 0.206·15-s − 0.856·17-s + 1.06·19-s − 1.30·23-s + 0.912·25-s − 0.00712·27-s − 0.845·29-s + 0.0926·31-s + 0.126·33-s − 1.01·37-s − 0.180·39-s + 1.05·41-s − 0.993·43-s + 0.0397·45-s − 2·49-s + 0.164·51-s + 0.528·53-s − 0.706·55-s − 0.204·57-s − 0.900·59-s + 0.495·61-s + 1.00·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1728\)    =    \(2^{6} \cdot 3^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 1728,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+3^{ -s})^{-1}\prod_{p \nmid 24 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 1.096127984\] \[L(1, E, \mathrm{sym}^{3}) \approx 1.086351677\]

Imaginary part of the first few zeros on the critical line

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