# Properties

 Degree 3 Conductor $2^{2} \cdot 5^{2} \cdot 73^{2} \cdot 137^{2}$ Sign $1$ Motivic weight 2 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{2})$  = 1 + 0.5·2-s + 0.333·3-s + 0.250·4-s + 0.200·5-s + 0.166·6-s + 1.28·7-s + 0.125·8-s − 0.222·9-s + 0.100·10-s + 0.454·11-s + 0.0833·12-s − 0.692·13-s + 0.642·14-s + 0.0666·15-s + 0.0625·16-s − 0.764·17-s − 0.111·18-s − 0.157·19-s + 0.0500·20-s + 0.428·21-s + 0.227·22-s − 0.304·23-s + 0.0416·24-s + 0.0400·25-s − 0.346·26-s + 0.814·27-s + 0.321·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut &\left(2^{2} \cdot 5^{2} \cdot 73^{2} \cdot 137^{2}\right)^{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

 $$d$$ = $$3$$ $$N$$ = $$2^{2} \cdot 5^{2} \cdot 73^{2} \cdot 137^{2}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(3,\ 2^{2} \cdot 5^{2} \cdot 73^{2} \cdot 137^{2} ,\ ( 1 : 1.0 ),\ 1 )$

## Euler product

\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-5^{- s})^{-1}(1-73^{- s})^{-1}(1-137^{- s})^{-1}\prod_{p \nmid 100010 }\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): 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.