Properties

Degree 3
Conductor $ 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  − 2-s − 0.666·3-s + 2·4-s + 0.200·5-s + 0.666·6-s + 0.142·7-s − 2·8-s + 1.11·9-s − 0.200·10-s − 0.181·11-s − 1.33·12-s + 0.923·13-s − 0.142·14-s − 0.133·15-s + 3·16-s − 0.470·17-s − 1.11·18-s − 0.789·19-s + 0.400·20-s − 0.0952·21-s + 0.181·22-s + 0.565·23-s + 1.33·24-s + 0.0400·25-s − 0.923·26-s − 0.185·27-s + 0.285·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(1225\)    =    \(5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 1225,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-5^{- s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 35 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.042613160\] \[L(1, E, \mathrm{sym}^{2}) \approx 0.8347393378\]

Imaginary part of the first few zeros on the critical line

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