Properties

Degree 3
Conductor 151321
Sign $1$
Self-dual yes
Motivic weight 2

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 + 0.800·5-s + 0.333·6-s + 2.571·7-s − 0.222·9-s + 0.800·10-s + 0.454·11-s − 0.307·13-s + 2.571·14-s + 0.266·15-s + 16-s + 1.117·17-s − 0.222·18-s + 0.315·19-s + 0.857·21-s + 0.454·22-s − 0.304·23-s − 0.160·25-s − 0.307·26-s + 0.814·27-s + 0.241·29-s + 0.266·30-s − 0.483·31-s + 32-s + 0.151·33-s + 1.117·34-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 151321 ^{s/2} \Gamma_{\R}(s+1) \Gamma_{\C}(s+1) \cdot L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{align} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(151321\)    =    \(389^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 151321,\ (1:1.0),\ 1)$

Euler product

\[\begin{equation} L(s, E, \mathrm{sym}^{2}) = (1-389^{- s})^{-1}\prod_{p \nmid 389 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{equation}\]

Particular Values

L(1/2): not computed L(1): not computed

Imaginary part of the first few zeros on the critical line

Zeros not available.

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

Plot not available.