# Properties

 Degree 3 Conductor $2^{2} \cdot 7^{2} \cdot 13^{2}$ Sign $1$ Motivic weight 2 Primitive yes Self-dual yes

# Learn more about

(not yet available)

## Dirichlet series

 $L(s, E, \mathrm{sym}^{2})$  = 1 + 0.5·2-s − 0.666·3-s + 0.250·4-s − 5-s − 0.333·6-s + 0.142·7-s + 0.125·8-s + 1.11·9-s − 0.5·10-s − 0.181·11-s − 0.166·12-s + 0.0769·13-s + 0.0714·14-s + 0.666·15-s + 0.0625·16-s − 17-s + 0.555·18-s − 0.789·19-s − 0.250·20-s − 0.0952·21-s − 0.0909·22-s − 0.608·23-s − 0.0833·24-s + 2·25-s + 0.0384·26-s − 0.185·27-s + 0.0357·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 33124 ^{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

 $$d$$ = $$3$$ $$N$$ = $$33124$$    =    $$2^{2} \cdot 7^{2} \cdot 13^{2}$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $(3,\ 33124,\ (1:1.0),\ 1)$

## Euler product

\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-7^{- s})^{-1}(1-13^{- s})^{-1}\prod_{p \nmid 182 }\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.