Properties

Degree 3
Conductor 4
Sign $-0.5 + 0.866i$
Self-dual no

Related objects

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Dirichlet series

$L(s,f)$  = 1  + (−0.25 − 0.433i) 2-s + (0.150 + 0.786i) 3-s + (−0.125 + 0.216i) 4-s + (1.159 − 0.148i) 5-s + (0.303 − 0.261i) 6-s + (−0.186 + 0.826i) 7-s +(0.125)·8-s + (−0.746 + 1.022i) 9-s + (−0.354 − 0.464i) 10-s + (0.166 − 0.581i) 11-s + (−0.189 − 0.065i) 12-s + (−0.265 + 0.005i) 13-s + (0.404 − 0.126i) 14-s + (0.291 + 0.889i) 15-s + (−0.031 − 0.054i) 16-s + (−0.711 − 1.254i) 17-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut & 4 ^{s/2} \Gamma_{\R}(s+9.632i) \Gamma_{\R}(s+1.374i) \Gamma_{\R}(s-11.00 i) \cdot L(s,f)\cr =\mathstrut & (-0.5 + 0.866i) \Lambda(1-s,\overline{f}) \end{align} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $-0.5 + 0.866i$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(3,\ 4,\ (9.63244453i, 1.3740602838i, -11.006504814i:\ ),\ -0.5 + 0.866i)$

Euler product

\[\begin{equation} L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \overline{a(p)} p^{-2s} - p^{-3s})^{-1} \end{equation}\]

Imaginary part of the first few zeros on the critical line

−24.827213, −23.759265, −22.556509, −20.850137, −19.503737, −17.825519, −17.197402, −14.699394, −13.175860, −6.562632, 2.532102, 5.347232, 9.173289, 13.859784, 16.054131, 17.551677, 19.159041, 20.684415, 21.753908, 22.426191, 24.786086

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