Properties

Degree 3
Conductor $ 3^{2} \cdot 5^{2} \cdot 7^{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  − 0.5·2-s + 0.333·3-s + 0.750·4-s + 0.200·5-s − 0.166·6-s + 0.142·7-s + 0.375·8-s + 0.111·9-s − 0.100·10-s − 11-s + 0.250·12-s + 1.76·13-s − 0.0714·14-s + 0.0666·15-s − 0.312·16-s − 0.764·17-s − 0.0555·18-s + 2.36·19-s + 0.149·20-s + 0.0476·21-s + 0.5·22-s + 1.78·23-s + 0.125·24-s + 0.0400·25-s − 0.884·26-s + 0.0370·27-s + 0.107·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 11025 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 11025,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1-5^{- s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 105 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.463599705\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.334330619\]

Imaginary part of the first few zeros on the critical line

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