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Properties

Label 1-1984-1984.395-r0-0-0
Degree $1$
Conductor $1984$
Sign $-0.265 - 0.964i$
Analytic cond. $9.21365$
Root an. cond. $9.21365$
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  + (−0.522 + 0.852i)3-s + (−0.382 + 0.923i)5-s + (0.453 − 0.891i)7-s + (−0.453 − 0.891i)9-s + (0.0784 − 0.996i)11-s + (−0.852 − 0.522i)13-s + (−0.587 − 0.809i)15-s + (0.951 + 0.309i)17-s + (−0.233 + 0.972i)19-s + (0.522 + 0.852i)21-s + (0.453 + 0.891i)23-s + (−0.707 − 0.707i)25-s + (0.996 + 0.0784i)27-s + (−0.972 − 0.233i)29-s + (0.809 + 0.587i)33-s + ⋯
L(s)  = 1  + (−0.522 + 0.852i)3-s + (−0.382 + 0.923i)5-s + (0.453 − 0.891i)7-s + (−0.453 − 0.891i)9-s + (0.0784 − 0.996i)11-s + (−0.852 − 0.522i)13-s + (−0.587 − 0.809i)15-s + (0.951 + 0.309i)17-s + (−0.233 + 0.972i)19-s + (0.522 + 0.852i)21-s + (0.453 + 0.891i)23-s + (−0.707 − 0.707i)25-s + (0.996 + 0.0784i)27-s + (−0.972 − 0.233i)29-s + (0.809 + 0.587i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $-0.265 - 0.964i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ -0.265 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1964153079 - 0.2577672485i\)
\(L(\frac12)\) \(\approx\) \(0.1964153079 - 0.2577672485i\)
\(L(1)\) \(\approx\) \(0.7013907173 + 0.1351979920i\)
\(L(1)\) \(\approx\) \(0.7013907173 + 0.1351979920i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (-0.522 + 0.852i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (0.453 - 0.891i)T \)
11 \( 1 + (0.0784 - 0.996i)T \)
13 \( 1 + (-0.852 - 0.522i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (-0.233 + 0.972i)T \)
23 \( 1 + (0.453 + 0.891i)T \)
29 \( 1 + (-0.972 - 0.233i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (0.987 - 0.156i)T \)
43 \( 1 + (-0.522 - 0.852i)T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (0.649 - 0.760i)T \)
59 \( 1 + (0.233 + 0.972i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + (-0.891 + 0.453i)T \)
73 \( 1 + (-0.891 - 0.453i)T \)
79 \( 1 + (0.951 + 0.309i)T \)
83 \( 1 + (0.233 - 0.972i)T \)
89 \( 1 + (-0.453 + 0.891i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−20.04809688750906841133924828965, −19.37665781744136541213806877860, −18.75598271518588050453220139393, −17.92460295051460745276648091221, −17.34938950067537820521877772466, −16.642804389148961916644576729663, −15.99377334965295222312248604081, −14.92335764208153244113823320683, −14.46892329245563605854302694739, −13.297073002223494145072027992917, −12.54215375258070647285908779106, −12.23973845996674287006830333502, −11.56659774897044150921098146613, −10.77829058682911266627231713392, −9.45727817146403575149943553714, −9.02804818208417810410077329585, −7.951782771638422094910923160621, −7.474218481162673090139794558622, −6.61067645336978999229713776747, −5.57149509342028179728490686684, −4.93694587491794008282998853180, −4.40407973307722062393716114164, −2.79568297626216620234108657571, −2.005951281152666436650198807885, −1.19597772208854789694539396628, 0.12819923992392595709643810942, 1.41392789567358983221879851251, 2.94799520774509801165032208333, 3.58399899588092721769299788951, 4.17594965248140103184184582285, 5.34263149427745501962043102361, 5.8578880714525583921034704871, 6.92657986741055687684470601208, 7.65714783900502299213531995776, 8.38891689980077135353904536628, 9.5791387686514287801081049329, 10.39508146965384199681025236225, 10.59773366254575259121018009759, 11.599176263454231023051107171442, 11.96324293494266489205802023933, 13.249916611744600502165905505, 14.11190607438543831902439414763, 14.83804570963136754000343008344, 15.13770242277570303729142427502, 16.32029727938580053044729291687, 16.75663537936550580484089826886, 17.437876002625831577815989910058, 18.22194831709866973599788319591, 19.11926884932139123576106146906, 19.67565312352108991198716168006

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