Properties

Degree 3
Conductor 4
Sign $1$
Self-dual no

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Dirichlet series

$L(s,f)$  = 1  +(0.5)·2-s + (1.066 + 0.481i) 3-s +(0.25)·4-s + (−0.248 − 0.072i) 5-s + (0.533 + 0.240i) 6-s + (0.676 − 0.291i) 7-s +(0.125)·8-s + (−0.161 + 1.506i) 9-s + (−0.124 − 0.036i) 10-s + (−0.264 − 0.294i) 11-s + (0.266 + 0.120i) 12-s + (−0.771 − 0.003i) 13-s + (0.338 − 0.145i) 14-s + (−0.230 − 0.196i) 15-s +(0.0625)·16-s + (0.335 − 0.143i) 17-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut & 4 ^{s/2} \Gamma_{\R}(s+8.239i) \Gamma_{\R}(s+2.641i) \Gamma_{\R}(s-10.88 i) \cdot L(s,f)\cr =\mathstrut & \Lambda(1-s,\overline{f}) \end{align} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(3,\ 4,\ (8.239796i, 2.641226i, -10.881024i:\ ),\ 1)$

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

−24.1020, −23.0757, −21.3176, −20.3333, −19.1542, −17.5728, −15.1645, −14.2232, −12.2527, 2.6576, 4.6967, 8.0717, 13.8334, 15.0792, 16.8507, 19.1905, 20.2680, 21.3171, 22.6026, 24.1266

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