Properties

Label 1-967-967.145-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.993 - 0.110i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (0.995 + 0.0909i)2-s + (−0.0682 − 0.997i)3-s + (0.983 + 0.181i)4-s + (−0.158 − 0.987i)5-s + (0.0227 − 0.999i)6-s + (−0.949 − 0.313i)7-s + (0.962 + 0.269i)8-s + (−0.990 + 0.136i)9-s + (−0.0682 − 0.997i)10-s + (−0.990 + 0.136i)11-s + (0.113 − 0.993i)12-s + (0.613 − 0.789i)13-s + (−0.917 − 0.398i)14-s + (−0.974 + 0.225i)15-s + (0.934 + 0.356i)16-s + (0.682 − 0.730i)17-s + ⋯
L(s)  = 1  + (0.995 + 0.0909i)2-s + (−0.0682 − 0.997i)3-s + (0.983 + 0.181i)4-s + (−0.158 − 0.987i)5-s + (0.0227 − 0.999i)6-s + (−0.949 − 0.313i)7-s + (0.962 + 0.269i)8-s + (−0.990 + 0.136i)9-s + (−0.0682 − 0.997i)10-s + (−0.990 + 0.136i)11-s + (0.113 − 0.993i)12-s + (0.613 − 0.789i)13-s + (−0.917 − 0.398i)14-s + (−0.974 + 0.225i)15-s + (0.934 + 0.356i)16-s + (0.682 − 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.993 - 0.110i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.993 - 0.110i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08709726331 - 1.564940790i\)
\(L(\frac12)\) \(\approx\) \(0.08709726331 - 1.564940790i\)
\(L(1)\) \(\approx\) \(1.153670792 - 0.8063498802i\)
\(L(1)\) \(\approx\) \(1.153670792 - 0.8063498802i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.995 + 0.0909i)T \)
3 \( 1 + (-0.0682 - 0.997i)T \)
5 \( 1 + (-0.158 - 0.987i)T \)
7 \( 1 + (-0.949 - 0.313i)T \)
11 \( 1 + (-0.990 + 0.136i)T \)
13 \( 1 + (0.613 - 0.789i)T \)
17 \( 1 + (0.682 - 0.730i)T \)
19 \( 1 + (-0.829 + 0.557i)T \)
23 \( 1 + (-0.334 - 0.942i)T \)
29 \( 1 + (-0.990 - 0.136i)T \)
31 \( 1 + (-0.715 + 0.699i)T \)
37 \( 1 + (0.613 - 0.789i)T \)
41 \( 1 + (0.854 + 0.519i)T \)
43 \( 1 + (0.291 - 0.956i)T \)
47 \( 1 + (-0.974 + 0.225i)T \)
53 \( 1 + (-0.648 + 0.761i)T \)
59 \( 1 + (-0.648 - 0.761i)T \)
61 \( 1 + (-0.877 + 0.480i)T \)
67 \( 1 + (0.962 - 0.269i)T \)
71 \( 1 + (-0.576 - 0.816i)T \)
73 \( 1 + (-0.648 - 0.761i)T \)
79 \( 1 + (-0.648 - 0.761i)T \)
83 \( 1 + (0.0227 - 0.999i)T \)
89 \( 1 + (-0.247 + 0.968i)T \)
97 \( 1 + T \)
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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

−21.972996836721191269846186253544, −21.555286150950043258258573744896, −20.945027573086579047861767004484, −19.8572472532500729665023403107, −19.18602865087283896133319995202, −18.43738262387643017491410174520, −17.060270019115230124826790421179, −16.239847263988474957218357389661, −15.64204919024183308558400122374, −15.03840405884297696808600461702, −14.34914388729766838530876913655, −13.390197329854781853417889927957, −12.68537627994433623681359657654, −11.40792495726754808240117458732, −11.09306734145060119471734047600, −10.166813928242414179733456316341, −9.522332205531102159496406755392, −8.17710258002926081395138478526, −7.11921932160199463012998335755, −6.04038341024205078667572916392, −5.75580731210971874346117100993, −4.4155409185522707653678425380, −3.61578498056855408222142273085, −3.03373937421571008369228022433, −2.09306335184443524960034777402, 0.43533246230291236371584305974, 1.70735007371778551603304220057, 2.786689734952925262096455493452, 3.63831930545753801421384894411, 4.80374645469946091166188262737, 5.74379917907896657749922184504, 6.22079277711698255727786230350, 7.507164350182552385516465032560, 7.84939551779951149671087444179, 8.98024864224844585628451433879, 10.30779539181057126729131601689, 11.10263314200506729304857506705, 12.30087863587342510237178899326, 12.69389154041843268279785679038, 13.138600159379059029064114179285, 13.92389702548753593136464116768, 14.896624107254248766415770441332, 16.050854120116285968266973718498, 16.357014333863229641177400648797, 17.23941382375543813054530893039, 18.34595228783662479400557884314, 19.1351523200328760137629329908, 20.12075129120813114431532189251, 20.439387069592582523019635890228, 21.27352690514554391806002292970

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