Properties

Degree 3
Conductor $ 2^{4} \cdot 3^{2} \cdot 139^{2} $
Sign $1$
Motivic weight 2
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  + 0.333·3-s + 0.800·5-s + 0.285·7-s + 0.111·9-s − 0.181·11-s − 0.307·13-s + 0.266·15-s + 1.11·17-s − 0.789·19-s + 0.0952·21-s + 1.78·23-s − 0.160·25-s + 0.0370·27-s − 0.137·29-s − 0.967·31-s − 0.0606·33-s + 0.228·35-s + 1.70·37-s − 0.102·39-s − 0.902·41-s − 0.162·43-s + 0.0888·45-s + 0.361·47-s − 0.204·49-s + 0.372·51-s − 0.698·53-s − 0.145·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 2782224 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(2782224\)    =    \(2^{4} \cdot 3^{2} \cdot 139^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 2782224,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1-139^{- s})^{-1}\prod_{p \nmid 10008 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.