# Properties

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

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## 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

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

## 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.