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Properties

Label 1-1984-1984.643-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: $\chi_{1984} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ 0.448 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.728665850 + 1.066892855i\)
\(L(\frac12)\) \(\approx\) \(1.728665850 + 1.066892855i\)
\(L(1)\) \(\approx\) \(1.374779472 + 0.2172924358i\)
\(L(1)\) \(\approx\) \(1.374779472 + 0.2172924358i\)

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

−19.651351182657676700035050374594, −19.4052323964128952807647070750, −18.66638395731504633093171195311, −17.540937775121635717577716149368, −16.81468059823002562755483509647, −16.01756062768220411655799281319, −15.690947256354751805405966234687, −14.59909329192268733455390190982, −13.91972170046236108979949826087, −13.29800304468654181304577620093, −12.63772287982457586175310406790, −11.895737142024914969626525914758, −10.81734485009752585848681293712, −9.838444863186335633326881075766, −9.21932582189193354377570533041, −8.76046954419865688730996679609, −8.16297457567819402231464838744, −6.8135881955167301959637182068, −6.30277381852917596799651365524, −5.25877631255794321631597352032, −4.08393663776881107369797524183, −3.905976936633421736013295993910, −2.28181874158743290459116030865, −2.191470713333017514070970912239, −0.60900100348354791622102751626, 1.215548638010055944621166237520, 2.31187217794758704164242785913, 2.873202271596064499044612178044, 3.76272179537772201911969025392, 4.441891645473374358601057913043, 5.93988764618207133649507175059, 6.62874217660533161300177999018, 7.20072701710476175800174574428, 7.96077270313575881315833705666, 8.966364623135513447512855576745, 9.80173236923801397030023987002, 10.13607276151518253807872060891, 11.02751531815089877386732283833, 12.24142329520019715267047549070, 12.86565689414306898950120248764, 13.56155905489375716060579962594, 14.24348268581107841268244471244, 14.91669864692608698731690963491, 15.48659609711687559526687925324, 16.33154632709702587596930959419, 17.54320555552638283529133309759, 17.8265345219280199493526470219, 18.81868140301436672678409538122, 19.406767872795659553126413687607, 20.10158673505286381918861058887

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