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Properties

Label 1-89-89.43-r1-0-0
Degree $1$
Conductor $89$
Sign $0.916 - 0.399i$
Analytic cond. $9.56437$
Root an. cond. $9.56437$
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  + (−0.142 + 0.989i)2-s + (−0.479 + 0.877i)3-s + (−0.959 − 0.281i)4-s + (0.909 + 0.415i)5-s + (−0.800 − 0.599i)6-s + (−0.349 − 0.936i)7-s + (0.415 − 0.909i)8-s + (−0.540 − 0.841i)9-s + (−0.540 + 0.841i)10-s + (−0.415 − 0.909i)11-s + (0.707 − 0.707i)12-s + (−0.877 − 0.479i)13-s + (0.977 − 0.212i)14-s + (−0.800 + 0.599i)15-s + (0.841 + 0.540i)16-s + (0.989 − 0.142i)17-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (−0.479 + 0.877i)3-s + (−0.959 − 0.281i)4-s + (0.909 + 0.415i)5-s + (−0.800 − 0.599i)6-s + (−0.349 − 0.936i)7-s + (0.415 − 0.909i)8-s + (−0.540 − 0.841i)9-s + (−0.540 + 0.841i)10-s + (−0.415 − 0.909i)11-s + (0.707 − 0.707i)12-s + (−0.877 − 0.479i)13-s + (0.977 − 0.212i)14-s + (−0.800 + 0.599i)15-s + (0.841 + 0.540i)16-s + (0.989 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.916 - 0.399i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ 0.916 - 0.399i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6833262844 - 0.1424656998i\)
\(L(\frac12)\) \(\approx\) \(0.6833262844 - 0.1424656998i\)
\(L(1)\) \(\approx\) \(0.6610062865 + 0.2901664000i\)
\(L(1)\) \(\approx\) \(0.6610062865 + 0.2901664000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.142 + 0.989i)T \)
3 \( 1 + (-0.479 + 0.877i)T \)
5 \( 1 + (0.909 + 0.415i)T \)
7 \( 1 + (-0.349 - 0.936i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (-0.877 - 0.479i)T \)
17 \( 1 + (0.989 - 0.142i)T \)
19 \( 1 + (-0.977 - 0.212i)T \)
23 \( 1 + (0.212 - 0.977i)T \)
29 \( 1 + (-0.936 + 0.349i)T \)
31 \( 1 + (0.212 + 0.977i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.877 - 0.479i)T \)
43 \( 1 + (-0.936 - 0.349i)T \)
47 \( 1 + (0.281 - 0.959i)T \)
53 \( 1 + (-0.281 - 0.959i)T \)
59 \( 1 + (0.479 + 0.877i)T \)
61 \( 1 + (-0.0713 - 0.997i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (-0.909 + 0.415i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (0.540 - 0.841i)T \)
83 \( 1 + (0.800 + 0.599i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−29.9008399463404547239914412579, −29.37086066718037616164120256282, −28.33660465729042841340241588092, −27.88771526353556908685113217764, −25.92974096049675754317677433828, −25.189513084491211488520806507329, −23.86156977683091894291771935725, −22.687246429490985544148026642619, −21.7485536107852984204479961981, −20.791644586774789678333800191287, −19.36177323355169819351790539494, −18.61991907526497684074575030862, −17.55303874225144308563886440506, −16.81525157174506612402455338522, −14.676627061073510072531411853053, −13.268775056704837895212632401128, −12.57222145812753073734378729140, −11.74834801942785875541551361218, −10.14879385006846971562274649132, −9.20961212074013980645818013582, −7.743598958490780687688669885880, −5.95442682094241201251221232647, −4.89248952859800155920483650440, −2.53853831760018571913528081634, −1.63938132971331054465427466045, 0.35722254344418695433849993381, 3.39470833777988596393188833640, 4.97540438778263784169062758135, 5.97127647066143116463379279974, 7.13396296843889518841089009473, 8.81963623209274013772291259395, 10.11975400438889093275751638829, 10.59682720657636462863408787774, 12.825397861670182960445889711747, 14.095368442890044917982197170978, 14.897735429597213434497271470992, 16.349117636173897769259698533977, 16.93003866380526104202133015281, 17.87095857300961113210634859878, 19.20260100063689711312247071724, 20.89924927934413075352544692143, 21.94263522952942554845505752146, 22.79275677445246550795734438754, 23.74959911779315276985235829976, 25.0946504278140050942073417733, 26.24341094752033797422470598744, 26.70694238390101929701499699096, 27.80520384218630053412809875613, 29.08971840701880936749369504951, 29.95184155344507223618021234825

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