Properties

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

Related objects

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Dirichlet series

$L(s,f)$  = 1  + (−0.421 + 1.067i) 2-s + (−0.768 − 1.313i) 3-s + (−0.541 + 0.167i) 4-s + (−0.400 + 0.239i) 5-s + (1.726 − 0.266i) 6-s + (−0.117 + 0.553i) 7-s + (−0.268 − 0.648i) 8-s + (−0.366 + 0.703i) 9-s + (−0.087 − 0.528i) 10-s + (−0.041 − 0.100i) 11-s + (0.635 + 0.582i) 12-s + (−0.309 − 0.328i) 13-s + (−0.541 − 0.358i) 14-s + (0.622 + 0.341i) 15-s + (−0.022 + 0.546i) 16-s + (0.259 − 0.620i) 17-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+16.40i)\Gamma_{\R}(s+0.171i)\Gamma_{\R}(s-16.57 i) \cdot L(s,f)\cr =\mathstrut & \Lambda(1-s,\overline{f}) \end{align} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(1\)    =    \(1\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(3,\ 1,\ (16.4031247403i, 0.171121891728i, -16.574246632i:\ ),\ 1)$

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

−23.4129437373605, −21.6868858848525, −19.8646960903853, −11.1407921358216, −9.8664332915609, −4.6144521141879, 6.4222353306131, 7.8655276953699, 12.3429586556897, 18.3920792539766, 22.6875461169111, 24.2616362328567

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