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Properties

Label 1-1984-1984.653-r0-0-0
Degree $1$
Conductor $1984$
Sign $-0.551 - 0.834i$
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  + (−0.760 − 0.649i)3-s + (0.923 − 0.382i)5-s + (0.156 − 0.987i)7-s + (0.156 + 0.987i)9-s + (0.852 + 0.522i)11-s + (0.649 − 0.760i)13-s + (−0.951 − 0.309i)15-s + (0.587 − 0.809i)17-s + (0.0784 − 0.996i)19-s + (−0.760 + 0.649i)21-s + (−0.156 − 0.987i)23-s + (0.707 − 0.707i)25-s + (0.522 − 0.852i)27-s + (−0.996 − 0.0784i)29-s + (−0.309 − 0.951i)33-s + ⋯
L(s)  = 1  + (−0.760 − 0.649i)3-s + (0.923 − 0.382i)5-s + (0.156 − 0.987i)7-s + (0.156 + 0.987i)9-s + (0.852 + 0.522i)11-s + (0.649 − 0.760i)13-s + (−0.951 − 0.309i)15-s + (0.587 − 0.809i)17-s + (0.0784 − 0.996i)19-s + (−0.760 + 0.649i)21-s + (−0.156 − 0.987i)23-s + (0.707 − 0.707i)25-s + (0.522 − 0.852i)27-s + (−0.996 − 0.0784i)29-s + (−0.309 − 0.951i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $-0.551 - 0.834i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ -0.551 - 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7705506386 - 1.433286204i\)
\(L(\frac12)\) \(\approx\) \(0.7705506386 - 1.433286204i\)
\(L(1)\) \(\approx\) \(0.9647656901 - 0.5340539361i\)
\(L(1)\) \(\approx\) \(0.9647656901 - 0.5340539361i\)

Euler product

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

−20.55701014795247278640061878716, −19.20382526884271525842059760103, −18.678458585175849413434511896894, −18.02663276224515607667366290001, −17.14175855898657244856303237186, −16.764348889025933479673158250198, −15.90211805100504903386016412930, −15.09356034623016222248207362084, −14.446983219160798009752233944993, −13.77531346557230972504105530489, −12.71015698497058003460550063919, −11.892978924273839010475903409443, −11.403484521411674575555143247335, −10.52540740186786176953556131300, −9.849746066500933433392404933096, −9.07056615212990874190823937902, −8.57387429005298032739004250853, −7.12940232514575516326841335145, −6.260390441185973470811911778439, −5.72125646563867826340824889280, −5.2927386507351128313386673658, −3.836995610578247835288529460788, −3.48592901210611212471392929571, −1.98317106400256711515858099636, −1.373305156154301875970115448941, 0.679431322765313613105024263465, 1.31385022920770509767125219462, 2.230184520516688319710117855167, 3.43973576860312934923273478217, 4.68037885449869454337313488322, 5.10865550860017547634052223087, 6.18826506996404401616838798215, 6.70095912825053855867566977637, 7.4941816156221567250624803072, 8.35086407275717771330059322597, 9.39902278487055455383715733388, 10.10121975930020839803895057956, 10.86550341383316886193659380028, 11.53121250643547525764278758484, 12.52441948181552376198258623744, 13.01502342565152197485503268817, 13.82378260901930012177999873593, 14.22588393831990763057996217943, 15.43719776417153992582605792040, 16.4222827240875741215596839665, 17.01746021947179242533001381044, 17.39579227984496891487351255751, 18.19417897052958710038757947525, 18.70596910615289926977756093034, 20.0146856749197179469800718274

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