Properties

Degree 4
Conductor $ 3^{4} \cdot 5^{4} \cdot 13^{4} $
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 + 0.375·4-s − 0.662·8-s + 0.657·11-s + 0.546·16-s − 0.856·17-s + 1.06·19-s − 0.697·22-s − 0.691·29-s + 0.580·32-s + 0.907·34-s − 1.15·37-s − 1.12·38-s + 0.685·41-s + 0.993·43-s + 0.246·44-s + 0.744·47-s − 2·49-s + 0.155·53-s + 0.733·58-s + 0.900·59-s + 0.495·61-s − 0.697·64-s + 0.218·67-s − 0.321·68-s + 1.04·71-s − 0.737·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3^{4} \cdot 5^{4} \cdot 13^{4}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1.5, 0.5 ),\ -1 )$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = \prod_{p \nmid 38025 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

Particular Values

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

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.