Properties

Degree 3
Conductor $ 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  + 2-s + 0.333·3-s − 5-s + 0.333·6-s + 0.285·7-s + 0.111·9-s − 10-s − 0.636·11-s − 0.923·13-s + 0.285·14-s − 0.333·15-s + 16-s − 0.764·17-s + 0.111·18-s + 0.315·19-s + 0.0952·21-s − 0.636·22-s + 0.565·23-s + 25-s − 0.923·26-s + 0.0370·27-s + 2.44·29-s − 0.333·30-s − 0.709·31-s + 32-s − 0.212·33-s − 0.764·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(225\)    =    \(3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 225,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1+5\ 5^{- s})^{-1}\prod_{p \nmid 75 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 1.325233443\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.371662535\]

Imaginary part of the first few zeros on the critical line

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