# Properties

 Degree $4$ Conductor $12678309000$ Sign $1$ Motivic weight $3$ Primitive yes Self-dual yes

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## Dirichlet series

 $L(s, E, \mathrm{sym}^{3})$  = 1 − 0.353·2-s − 0.192·3-s + 0.125·4-s + 0.0894·5-s + 0.0680·6-s − 0.0539·7-s − 0.0441·8-s + 0.0370·9-s − 0.0316·10-s − 2.30·11-s − 0.0240·12-s − 1.28·13-s + 0.0190·14-s − 0.0172·15-s + 0.0156·16-s − 0.171·17-s − 0.0130·18-s + 0.0111·20-s + 0.0103·21-s + 0.814·22-s − 0.761·23-s + 0.00850·24-s + 0.00800·25-s + 0.452·26-s − 0.00712·27-s − 0.00674·28-s − 0.941·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 37^{2}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 37^{2}$$ Sign: $1$ Primitive: yes Self-dual: yes Selberg data: $$(4,\ 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 37^{2} ,\ ( \ : 1.5, 0.5 ),\ 1 )$$

## Particular Values

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

## Euler product

$$L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1}(1+3^{ -s})^{-1}(1-5^{- s})^{-1}(1+7^{ -s})^{-1}(1+110\ 37^{- s}+50653\ 37^{-2 s})^{-1}\prod_{p \nmid 287490 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}$$

## 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.