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Properties

Label 1-1984-1984.523-r0-0-0
Degree $1$
Conductor $1984$
Sign $0.448 - 0.893i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $0.448 - 0.893i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ 0.448 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1970054312 - 0.1215872269i\)
\(L(\frac12)\) \(\approx\) \(0.1970054312 - 0.1215872269i\)
\(L(1)\) \(\approx\) \(0.5171868337 + 0.1015140346i\)
\(L(1)\) \(\approx\) \(0.5171868337 + 0.1015140346i\)

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

−20.07529754673910118720841222481, −19.41955878663240069042284672628, −18.58999455164881178641531924511, −17.85024962910237396889325600045, −16.95317313535532776215200554959, −16.45630597474572786304523242074, −15.77255899383961610484878739074, −15.59019315709730809744597833167, −13.94942156260535559158031504475, −13.34060084135709559142960903767, −12.77607266870062444058246413499, −11.78128740849245268152534339077, −11.38179278127910434426726173175, −10.53832491678521530961640194678, −9.51052982121709946927115054710, −9.18786181305175537742113677255, −7.99094544616745764240361310291, −7.174459679688027420104405131998, −6.31770866106823241124197323517, −5.63638845728903833698307269386, −4.73222814208246172386022901778, −4.087837085693000213045092267935, −3.26216475427727864627947103735, −1.81787185231484020668282450856, −0.66312934430166547831301372308, 0.14254539806662162050308997113, 1.7246929647277134632607248074, 2.69636819159406722658022085339, 3.53571261096335232274129252219, 4.53071579623814715654065224309, 5.57547359731265618584882306893, 6.10547867772872224021236206277, 6.978394157814715116765391112499, 7.495142137247291040918282733237, 8.45735472758084466220241937057, 9.700170671730645304610922948522, 10.39355348941924495978650323636, 10.75063214787953057076818204550, 11.87152232272536543996943859419, 12.31293025076632156041993794383, 13.083988547668936325155795904169, 13.801448510747600576320975531238, 15.07048487458643622227683832332, 15.51372019140597559370704768093, 16.05796663528620177133080763340, 16.97006858077469174844635541069, 17.94494348608456425525238389205, 18.3094720266523758122877251666, 18.79866116571904169363046283234, 19.87015143902601848315788108638

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