Properties

Degree 3
Conductor $ 2^{4} \cdot 13^{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  − 0.666·3-s − 0.800·5-s + 2.57·7-s + 1.11·9-s − 0.636·11-s + 0.0769·13-s + 0.533·15-s − 0.470·17-s − 0.789·19-s − 1.71·21-s − 0.304·23-s + 1.43·25-s − 0.185·27-s + 0.241·29-s − 0.483·31-s + 0.424·33-s − 2.05·35-s + 2.27·37-s − 0.0512·39-s + 0.560·41-s − 0.976·43-s − 0.888·45-s + 0.723·47-s + 4.04·49-s + 0.313·51-s + 1.71·53-s + 0.509·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 2704 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(2704\)    =    \(2^{4} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 2704,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-13^{- s})^{-1}\prod_{p \nmid 104 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.502337037\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.060765611\]

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line