Properties

Degree 4
Conductor $ 17^{3} $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes

Related objects

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Normalization:  

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Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  + 1.06·2-s + 0.375·4-s + 1.07·5-s + 0.431·7-s + 0.662·8-s − 2·9-s + 1.13·10-s + 0.938·13-s + 0.458·14-s + 0.546·16-s + 0.0142·17-s − 2.12·18-s + 1.06·19-s + 0.402·20-s − 1.08·23-s + 0.912·25-s + 0.995·26-s + 0.161·28-s − 0.845·29-s − 1.06·31-s − 0.580·32-s + 0.0151·34-s + 0.463·35-s − 0.750·36-s + 0.622·37-s + 1.12·38-s + 0.711·40-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 4913 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4913\)    =    \(17^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 4913,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-17^{- s})^{-1}\prod_{p \nmid 17 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 2.398566220\] \[L(1, E, \mathrm{sym}^{3}) \approx 1.895520176\]

Imaginary part of the first few zeros on the critical line

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