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Properties

Label 1-1984-1984.109-r0-0-0
Degree $1$
Conductor $1984$
Sign $-0.925 - 0.378i$
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  + (−0.996 − 0.0784i)3-s + (−0.923 + 0.382i)5-s + (0.987 − 0.156i)7-s + (0.987 + 0.156i)9-s + (0.233 − 0.972i)11-s + (0.0784 − 0.996i)13-s + (0.951 − 0.309i)15-s + (−0.587 − 0.809i)17-s + (0.649 − 0.760i)19-s + (−0.996 + 0.0784i)21-s + (−0.987 − 0.156i)23-s + (0.707 − 0.707i)25-s + (−0.972 − 0.233i)27-s + (−0.760 − 0.649i)29-s + (−0.309 + 0.951i)33-s + ⋯
L(s)  = 1  + (−0.996 − 0.0784i)3-s + (−0.923 + 0.382i)5-s + (0.987 − 0.156i)7-s + (0.987 + 0.156i)9-s + (0.233 − 0.972i)11-s + (0.0784 − 0.996i)13-s + (0.951 − 0.309i)15-s + (−0.587 − 0.809i)17-s + (0.649 − 0.760i)19-s + (−0.996 + 0.0784i)21-s + (−0.987 − 0.156i)23-s + (0.707 − 0.707i)25-s + (−0.972 − 0.233i)27-s + (−0.760 − 0.649i)29-s + (−0.309 + 0.951i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $-0.925 - 0.378i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ -0.925 - 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1083480649 - 0.5516415988i\)
\(L(\frac12)\) \(\approx\) \(0.1083480649 - 0.5516415988i\)
\(L(1)\) \(\approx\) \(0.6522222869 - 0.1814723736i\)
\(L(1)\) \(\approx\) \(0.6522222869 - 0.1814723736i\)

Euler product

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

−20.27648294456278652519243189745, −19.66216869282848287488322048225, −18.55082794428280059477821931161, −18.16986834560633832638462198131, −17.316565455762842333742510773311, −16.635618573995895886313602615654, −16.07973980377034252878362703293, −15.14349322569260113381242364256, −14.73419790769227448952253872935, −13.62985550960721683358700456020, −12.517491151917098883922651697527, −12.157943033042768029979942408137, −11.394744348873139394943722315640, −10.98015641591606494180285258419, −9.89775517067878731120340324536, −9.14618572668672516449295245356, −8.08003319485416282650171817499, −7.53355787896140441186782336292, −6.64695234022173389107990564171, −5.79043502741415716782706938333, −4.73980944921319313660498542843, −4.4269192927214259075797541390, −3.61318102103176102659414507348, −1.82967238175009111547962171819, −1.4146777808176756929180441568, 0.26291441613775614548669710307, 1.077712874762191281496524087111, 2.424226697728057182168303653673, 3.506830873087696926723410924512, 4.36479723925714305282000824876, 5.08387936691547556288375788823, 5.90581416191094314559056553089, 6.75650242533282534094228781458, 7.692128499642442044486498794191, 8.021962873058309263908631553774, 9.18165226324476103416631032053, 10.24302650602329174407006791529, 11.131968638722038476657527636884, 11.317218295098075155832876577221, 11.96117245418032941877059025188, 12.96123823952132669396932760484, 13.73810124998186145737604889795, 14.608264261248833456119396258740, 15.46260649343570282502882798526, 15.99721159983153691791638973872, 16.68316977212093211863454057043, 17.68954692941202032637933604547, 18.06718558610047966629887544906, 18.72268972158781300474875483315, 19.67681044972216485371829309345

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