Properties

Degree $1$
Conductor $89$
Sign $0.916 - 0.399i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.916 - 0.399i$
Motivic weight: \(0\)
Character: $\chi_{89} (43, \cdot )$
Sato-Tate group: $\mu(88)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ 0.916 - 0.399i)\)

Particular Values

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

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

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