# Properties

 Degree 3 Conductor $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2} \cdot 17^{2}$ 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 − 0.428·7-s + 0.125·8-s + 0.111·9-s + 0.100·10-s − 11-s + 0.0833·12-s + 0.0769·13-s − 0.214·14-s + 0.0666·15-s + 0.0625·16-s + 0.0588·17-s + 0.0555·18-s − 0.789·19-s + 0.0500·20-s − 0.142·21-s − 0.5·22-s + 1.78·23-s + 0.0416·24-s + 0.0400·25-s + 0.0384·26-s + 0.0370·27-s − 0.107·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 43956900 ^{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

 $$d$$ = $$3$$ $$N$$ = $$43956900$$    =    $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2} \cdot 17^{2}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $$(3,\ 43956900,\ (1:1.0),\ 1)$$

## Euler product

\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-5^{- s})^{-1}(1+3\ 7^{- s}-21 \ 7^{-2 s}-343 \ 7^{-3 s})^{-1}(1-13^{- s})^{-1}(1-17^{- s})^{-1}\prod_{p \nmid 324870 }\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.