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Properties

Label 1-1984-1984.349-r0-0-0
Degree $1$
Conductor $1984$
Sign $0.635 + 0.772i$
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  + (−0.972 + 0.233i)3-s + (−0.382 − 0.923i)5-s + (0.891 + 0.453i)7-s + (0.891 − 0.453i)9-s + (−0.649 − 0.760i)11-s + (−0.233 − 0.972i)13-s + (0.587 + 0.809i)15-s + (0.951 + 0.309i)17-s + (−0.852 + 0.522i)19-s + (−0.972 − 0.233i)21-s + (−0.891 + 0.453i)23-s + (−0.707 + 0.707i)25-s + (−0.760 + 0.649i)27-s + (0.522 + 0.852i)29-s + (0.809 + 0.587i)33-s + ⋯
L(s)  = 1  + (−0.972 + 0.233i)3-s + (−0.382 − 0.923i)5-s + (0.891 + 0.453i)7-s + (0.891 − 0.453i)9-s + (−0.649 − 0.760i)11-s + (−0.233 − 0.972i)13-s + (0.587 + 0.809i)15-s + (0.951 + 0.309i)17-s + (−0.852 + 0.522i)19-s + (−0.972 − 0.233i)21-s + (−0.891 + 0.453i)23-s + (−0.707 + 0.707i)25-s + (−0.760 + 0.649i)27-s + (0.522 + 0.852i)29-s + (0.809 + 0.587i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $0.635 + 0.772i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ 0.635 + 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7303205192 + 0.3447304256i\)
\(L(\frac12)\) \(\approx\) \(0.7303205192 + 0.3447304256i\)
\(L(1)\) \(\approx\) \(0.7391503863 + 0.002160168967i\)
\(L(1)\) \(\approx\) \(0.7391503863 + 0.002160168967i\)

Euler product

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

−19.673530768490500493602757608655, −18.97263761291291481030710076277, −18.32756911119234454924512397534, −17.7037503635663455363935870167, −17.154985405569993668838774393266, −16.22183969257831461386996396072, −15.60472758317998738352382479328, −14.60298581597971521168270643029, −14.20533104989819043163806356728, −13.16265191805390219797428633642, −12.295834279919656105332929229210, −11.5895026305268053261478331940, −11.12882627136962815735614113427, −10.244415830639351008724396281308, −9.8529028425697432207212275428, −8.31279410157001923853735354262, −7.61058525178349694504632243269, −7.01257692627944112726745960993, −6.34198864262640057484572639616, −5.30197729546859888677831990653, −4.50205459115689442002740617173, −3.92841038755555277693080956487, −2.42504926215155962811606215944, −1.82180550463716382226714519338, −0.402748237447832981148572554571, 0.89224245048106190595438145174, 1.6864038710094804835998307360, 3.10576536471851691855577866927, 4.10261233553305680888037493120, 4.93430562277966269009301968164, 5.52136222405364031051608352737, 6.016425291552409298782822559546, 7.373019138009025709915337488039, 8.24022568609255000265029204789, 8.51607301473383524099575233074, 9.9222547693881736462908358784, 10.364934952941841137120314627211, 11.387690728900935646986796247053, 11.83653777777647715361065605738, 12.64324998818062189926704643054, 13.12331660826728218052697321020, 14.33420630172484620682476224336, 15.19832052469431464764858091398, 15.73332948965552445500044970371, 16.61014272037042343620198709157, 16.95938989076938890465016970243, 17.93625485664599844017370351958, 18.380909609045499073965744663786, 19.28759898129887639893139226792, 20.231400852094710326188826761403

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