Properties

Degree 3
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{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 + 0.200·5-s + 1.28·7-s + 0.111·9-s − 11-s + 1.76·13-s + 0.0666·15-s − 0.764·17-s − 0.157·19-s + 0.428·21-s + 1.78·23-s + 0.0400·25-s + 0.0370·27-s + 0.241·29-s − 31-s − 0.333·33-s + 0.257·35-s − 0.0270·37-s + 0.589·39-s + 1.43·41-s − 0.627·43-s + 0.0222·45-s + 0.361·47-s + 0.367·49-s − 0.254·51-s + 0.886·53-s − 0.200·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 3600,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1-5^{- s})^{-1}\prod_{p \nmid 120 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.287660382\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.458363704\]

Imaginary part of the first few zeros on the critical line

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