Properties

Degree 3
Conductor $ 2^{4} \cdot 19^{2} $
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 − 0.800·5-s + 0.285·7-s − 0.222·9-s − 0.181·11-s + 0.230·13-s − 0.266·15-s + 0.470·17-s + 0.0526·19-s + 0.0952·21-s − 23-s + 1.43·25-s + 0.814·27-s − 0.862·29-s + 1.06·31-s − 0.0606·33-s − 0.228·35-s + 1.70·37-s + 0.0769·39-s − 0.121·41-s + 0.139·43-s + 0.177·45-s + 0.723·47-s − 0.204·49-s + 0.156·51-s + 0.207·53-s + 0.145·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 5776 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(5776\)    =    \(2^{4} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 5776,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-19^{- s})^{-1}\prod_{p \nmid 152 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.715738927\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.123440933\]

Imaginary part of the first few zeros on the critical line

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