Properties

Degree 4
Conductor $ 3^{3} \cdot 5^{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  + 1.06·2-s − 0.192·3-s + 0.375·4-s + 0.0894·5-s − 0.204·6-s + 0.662·8-s + 0.0370·9-s + 0.0948·10-s + 0.657·11-s − 0.0721·12-s + 0.938·13-s − 0.0172·15-s + 0.546·16-s − 0.856·17-s + 0.0392·18-s − 1.06·19-s + 0.0335·20-s + 0.697·22-s − 0.127·24-s + 0.00800·25-s + 0.995·26-s − 0.00712·27-s + 0.691·29-s − 0.0182·30-s − 0.580·32-s − 0.126·33-s − 0.907·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3375\)    =    \(3^{3} \cdot 5^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 3375,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+3^{ -s})^{-1}(1-5^{- s})^{-1}\prod_{p \nmid 15 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 1.934150652\] \[L(1, E, \mathrm{sym}^{3}) \approx 1.656256181\]

Imaginary part of the first few zeros on the critical line

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