Properties

Degree 3
Conductor $ 2^{8} $
Sign $1$
Motivic weight 2
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  + 0.333·3-s − 0.200·5-s + 1.28·7-s − 0.222·9-s − 0.636·11-s − 0.692·13-s − 0.0666·15-s − 0.764·17-s − 0.789·19-s + 0.428·21-s − 0.304·23-s + 0.239·25-s + 0.814·27-s + 0.241·29-s − 31-s − 0.212·33-s − 0.257·35-s + 1.70·37-s − 0.230·39-s − 0.121·41-s − 0.162·43-s + 0.0444·45-s + 0.361·47-s + 0.367·49-s − 0.254·51-s − 0.320·53-s + 0.127·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 256 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(256\)    =    \(2^{8}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 256,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = \prod_{p \nmid 128 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.066665237\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.088717758\]

Imaginary part of the first few zeros on the critical line

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