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Properties

Label 1-373-373.30-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.368 + 0.929i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.820 + 0.571i)2-s + (0.979 + 0.201i)3-s + (0.347 + 0.937i)4-s + (−0.612 + 0.790i)5-s + (0.688 + 0.724i)6-s + (0.528 + 0.848i)7-s + (−0.250 + 0.968i)8-s + (0.918 + 0.394i)9-s + (−0.954 + 0.299i)10-s + (−0.250 − 0.968i)11-s + (0.151 + 0.988i)12-s + (−0.250 − 0.968i)13-s + (−0.0506 + 0.998i)14-s + (−0.758 + 0.651i)15-s + (−0.758 + 0.651i)16-s + (0.347 + 0.937i)17-s + ⋯
L(s)  = 1  + (0.820 + 0.571i)2-s + (0.979 + 0.201i)3-s + (0.347 + 0.937i)4-s + (−0.612 + 0.790i)5-s + (0.688 + 0.724i)6-s + (0.528 + 0.848i)7-s + (−0.250 + 0.968i)8-s + (0.918 + 0.394i)9-s + (−0.954 + 0.299i)10-s + (−0.250 − 0.968i)11-s + (0.151 + 0.988i)12-s + (−0.250 − 0.968i)13-s + (−0.0506 + 0.998i)14-s + (−0.758 + 0.651i)15-s + (−0.758 + 0.651i)16-s + (0.347 + 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.368 + 0.929i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.368 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.485800794 + 2.186867558i\)
\(L(\frac12)\) \(\approx\) \(1.485800794 + 2.186867558i\)
\(L(1)\) \(\approx\) \(1.639792648 + 1.211899396i\)
\(L(1)\) \(\approx\) \(1.639792648 + 1.211899396i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.820 + 0.571i)T \)
3 \( 1 + (0.979 + 0.201i)T \)
5 \( 1 + (-0.612 + 0.790i)T \)
7 \( 1 + (0.528 + 0.848i)T \)
11 \( 1 + (-0.250 - 0.968i)T \)
13 \( 1 + (-0.250 - 0.968i)T \)
17 \( 1 + (0.347 + 0.937i)T \)
19 \( 1 + (-0.758 - 0.651i)T \)
23 \( 1 + (0.979 - 0.201i)T \)
29 \( 1 + (-0.994 + 0.101i)T \)
31 \( 1 + (0.151 - 0.988i)T \)
37 \( 1 + (-0.874 - 0.485i)T \)
41 \( 1 + (0.688 + 0.724i)T \)
43 \( 1 + (0.347 + 0.937i)T \)
47 \( 1 + (-0.0506 - 0.998i)T \)
53 \( 1 + (0.918 - 0.394i)T \)
59 \( 1 + (-0.994 + 0.101i)T \)
61 \( 1 + (0.979 + 0.201i)T \)
67 \( 1 + (-0.612 + 0.790i)T \)
71 \( 1 + (0.347 - 0.937i)T \)
73 \( 1 + (0.918 - 0.394i)T \)
79 \( 1 + (-0.954 + 0.299i)T \)
83 \( 1 + (0.918 - 0.394i)T \)
89 \( 1 + T \)
97 \( 1 + (0.347 + 0.937i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.20990324733780396954188426194, −23.48444145049287254985677811365, −22.842380318298528017069801073481, −21.23750170710921754490585659613, −20.78261212970243661537560945464, −20.22624926743065026699263266922, −19.34515124693785651007692608105, −18.66545240720080155885374978398, −17.20037707114979593384643548028, −16.06481747268846890454910663897, −15.14274541511858828119159390561, −14.34630669357820683979875684549, −13.60869730689520509811404779407, −12.648026454490970423611503760358, −12.01903634656528501427005998895, −10.84218744367222257968852131024, −9.74154305264507774774825913332, −8.87565930656699858263075506775, −7.52458161752747271630854077908, −6.97293228308156589605471553407, −5.066305689756728938983434077217, −4.35466980809697363376342997336, −3.57289256968799813928675881064, −2.14256506436648320222763254010, −1.223280858663125709952102563686, 2.32342467984404992455716426016, 3.071128895402144843928753681352, 3.95464773602406434464819386474, 5.169904218373181746059018934765, 6.24293617417162556634416797807, 7.50627954295374571405866047167, 8.14826755127632791421106771015, 8.90212803260260920300951981660, 10.59532851346038590950512246829, 11.35068722648810635259922640285, 12.61694106732507593556010757140, 13.33376971500746123970009407566, 14.5483180109155528272659096926, 15.005493927848041640766764644385, 15.48576875101158571657682425862, 16.592446410565143956334135799091, 17.86676581817453697419325807684, 18.88249419172186676409841954629, 19.604259632509470860468090655984, 20.82152975930284741500408890810, 21.50820929876599930736432324054, 22.15562721863727957579031473364, 23.16664414964600549524345501874, 24.24737341141394127840717261517, 24.68810475627408846917035001610

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