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Properties

Label 1-89-89.68-r0-0-0
Degree $1$
Conductor $89$
Sign $0.970 - 0.240i$
Analytic cond. $0.413314$
Root an. cond. $0.413314$
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  + (0.841 − 0.540i)2-s + (−0.909 + 0.415i)3-s + (0.415 − 0.909i)4-s + (0.142 + 0.989i)5-s + (−0.540 + 0.841i)6-s + (0.989 − 0.142i)7-s + (−0.142 − 0.989i)8-s + (0.654 − 0.755i)9-s + (0.654 + 0.755i)10-s + (−0.142 + 0.989i)11-s + i·12-s + (0.909 − 0.415i)13-s + (0.755 − 0.654i)14-s + (−0.540 − 0.841i)15-s + (−0.654 − 0.755i)16-s + (−0.841 − 0.540i)17-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.909 + 0.415i)3-s + (0.415 − 0.909i)4-s + (0.142 + 0.989i)5-s + (−0.540 + 0.841i)6-s + (0.989 − 0.142i)7-s + (−0.142 − 0.989i)8-s + (0.654 − 0.755i)9-s + (0.654 + 0.755i)10-s + (−0.142 + 0.989i)11-s + i·12-s + (0.909 − 0.415i)13-s + (0.755 − 0.654i)14-s + (−0.540 − 0.841i)15-s + (−0.654 − 0.755i)16-s + (−0.841 − 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.970 - 0.240i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ 0.970 - 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.297568431 - 0.1585428672i\)
\(L(\frac12)\) \(\approx\) \(1.297568431 - 0.1585428672i\)
\(L(1)\) \(\approx\) \(1.325384554 - 0.1531395497i\)
\(L(1)\) \(\approx\) \(1.325384554 - 0.1531395497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 + (-0.909 + 0.415i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (0.989 - 0.142i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (0.909 - 0.415i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (0.755 + 0.654i)T \)
23 \( 1 + (-0.755 - 0.654i)T \)
29 \( 1 + (-0.989 + 0.142i)T \)
31 \( 1 + (-0.755 + 0.654i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.909 + 0.415i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (-0.415 + 0.909i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (-0.909 - 0.415i)T \)
61 \( 1 + (0.281 - 0.959i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (-0.540 + 0.841i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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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

−30.67336597286210438482884019875, −29.62522936940789540505256671394, −28.588115151962208995036228466492, −27.60944467266535678330023214340, −26.16798751710146270209074549971, −24.64786362866589459974776970355, −24.14923737345556908597716329789, −23.54967129324110145633719456385, −22.01690360145144917457733221397, −21.370759973850384142146739718929, −20.18767151514304409720482751838, −18.33095820198204169906915713151, −17.32369071075956679867531813250, −16.43889595883247191458161416608, −15.507880689912102733318014375448, −13.74135748680493387862723968905, −13.145525140960653123622294417408, −11.73885409644561436903556178542, −11.15745408690458147442225108668, −8.74654842368673162147498152919, −7.68606338092277609039861809464, −6.09355380134506530873819871760, −5.301055349992012002516636383588, −4.141894370017551150138922315482, −1.70542595603129096585048369841, 1.844042923928456810025617633, 3.68297883346280854465951803985, 4.89945784775001896204941550323, 6.0597833922567588497686592916, 7.29433667304200971269522822836, 9.73733625206982259690442707212, 10.81385145359169529678941628213, 11.37563744313303320595235301555, 12.646543767785333515125698116020, 14.11956137974996145346663636036, 15.03631072741793997874383714197, 16.048094217413370555800594607889, 17.93493148358904549652355744542, 18.29396679473448698340469748383, 20.24040281263952843820583182832, 21.02259488123052884482563058425, 22.18933520501854735877236404364, 22.81937872336959057240542740489, 23.68945101367071454351350273979, 24.930703205935554162723737058617, 26.52206641292651457547097914538, 27.64630288142604012685502027931, 28.4759043200987680154371239036, 29.602324293842217359943461188405, 30.45521799832758911131872802086

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