Properties

Degree 4
Level 7112448
Sign $1$
Self-dual
Motivic weight 3

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Dirichlet series

$L(s, E, \mathrm{sym}^3) = 1^{\mathstrut}$  − 1.073 5-s + 0.053 7-s − 0.657 11-s + 0.938 13-s + 0.171 17-s + 1.062 19-s + 0.912 25-s − 0.691 29-s − 0.057 35-s − 1.013 37-s + 0.594 41-s − 0.993 43-s + 0.002 49-s + 1.088 53-s + 0.706 55-s + 0.688 59-s + 0.495 61-s − 1.007 65-s + 0.860 67-s + 1.058 73-s − 0.035 77-s + 2.233 79-s + 0.349 83-s − 0.183 85-s + 0.300 89-s + 0.050 91-s − 1.140 95-s + ...

Functional equation

\[\begin{align} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 7112448 ^{s/2} \Gamma_{\C}(s+1.5) \Gamma_{\C}(s+0.5) \cdot L(s, E, \mathrm{sym}^3)\cr =\mathstrut & \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{align} \]
Selberg data: $(4,7112448,(:1.5, 0.5), 1)$

Euler product

\[\begin{equation} L(s, E, \mathrm{sym}^3) = (1-7^{- s})^{-1}\prod_{p \nmid 1008 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{equation}\]

Imaginary part of the first few zeros on the critical line

Particular Values

\[L(1/2, E, \mathrm{sym}^3) \approx 2.2905238705\]
\[L(1, E, \mathrm{sym}^3) \approx 0.9617090133\]

Graph of the $Z$-function along center part of the critical line