Properties

Degree 4
Conductor $ 43^{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  + 0.769·3-s − 2.14·5-s + 0.814·9-s − 1.06·11-s + 0.106·13-s − 1.65·15-s − 16-s + 1.07·17-s + 0.821·19-s + 0.407·23-s + 1.96·25-s + 1.56·27-s + 0.845·29-s + 0.353·31-s − 0.822·33-s + 0.0821·39-s − 1.08·41-s − 0.00354·43-s − 1.74·45-s − 0.968·47-s − 0.769·48-s − 2·49-s + 0.823·51-s + 1.04·53-s + 2.29·55-s + 0.632·57-s − 0.688·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 79507 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(79507\)    =    \(43^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 79507,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+43^{ -s})^{-1}\prod_{p \nmid 43 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.