Properties

Degree 4
Conductor $ 61^{3} $
Sign $-1$
Motivic weight 3
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  + 1.06·2-s + 0.769·3-s + 0.375·4-s + 0.268·5-s + 0.816·6-s − 0.701·7-s + 0.662·8-s + 0.814·9-s + 0.284·10-s − 0.411·11-s + 0.288·12-s − 0.533·13-s − 0.744·14-s + 0.206·15-s + 0.546·16-s − 1.02·17-s + 0.864·18-s + 1.06·19-s + 0.100·20-s − 0.540·21-s − 0.436·22-s − 2.85·23-s + 0.510·24-s + 0.232·25-s − 0.565·26-s + 1.56·27-s − 0.263·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 226981 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(226981\)    =    \(61^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 226981,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+61^{ -s})^{-1}\prod_{p \nmid 61 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.