Properties

Degree $4$
Conductor $16875$
Sign $1$
Motivic weight $3$
Arithmetic yes
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 − 0.204·6-s − 0.662·8-s + 0.0370·9-s + 0.657·11-s + 0.0721·12-s − 0.938·13-s + 0.546·16-s + 0.856·17-s − 0.0392·18-s − 1.06·19-s − 0.697·22-s − 0.127·24-s + 0.995·26-s + 0.00712·27-s + 0.691·29-s + 0.580·32-s + 0.126·33-s − 0.907·34-s + 0.0138·36-s + 1.15·37-s + 1.12·38-s − 0.180·39-s + 0.685·41-s + 0.993·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 16875 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16875\)    =    \(3^{3} \cdot 5^{4}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 16875,\ (\ :1.5, 0.5),\ 1)\)

Particular Values

L(1/2): not computed L(1): not computed

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1} \prod_{p \nmid 75 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.