# Properties

 Degree $3$ Conductor $2690496900$ 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 + 0.142·7-s + 0.125·8-s + 0.111·9-s + 0.100·10-s − 11-s + 0.0833·12-s + 0.0769·13-s + 0.0714·14-s + 0.0666·15-s + 0.0625·16-s + 1.11·17-s + 0.0555·18-s + 0.0526·19-s + 0.0500·20-s + 0.0476·21-s − 0.5·22-s − 23-s + 0.0416·24-s + 0.0400·25-s + 0.0384·26-s + 0.0370·27-s + 0.0357·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut &\left(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{2} \cdot 19^{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

 Degree: $$3$$ Conductor: $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{2} \cdot 19^{2}$$ Sign: $1$ Primitive: yes Self-dual: yes Selberg data: $$(3,\ 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{2} \cdot 19^{2} ,\ ( 1 : 1.0 ),\ 1 )$$

## Particular Values

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

## Euler product

$$L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-5^{- s})^{-1}(1-7^{- s})^{-1}(1-13^{- s})^{-1}(1-19^{- s})^{-1}\prod_{p \nmid 51870 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}$$

## 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.