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Properties

Label 1-89-89.17-r0-0-0
Degree $1$
Conductor $89$
Sign $0.660 + 0.750i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.660 + 0.750i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ 0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.633007101 + 0.7382562632i\)
\(L(\frac12)\) \(\approx\) \(1.633007101 + 0.7382562632i\)
\(L(1)\) \(\approx\) \(1.685289393 + 0.5697825519i\)
\(L(1)\) \(\approx\) \(1.685289393 + 0.5697825519i\)

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

−30.554870952471922096526282533645, −29.504004262289860379175022853375, −28.85821566603287548977270792786, −27.13023626706893155706560701848, −25.89249729973020582732093451176, −25.19677629400853146199432237620, −23.903371284641974179462038115749, −22.78020517657791755158535618604, −21.952849900626556127718665103235, −20.77405601933705688908335882384, −19.4845932819693584446985178615, −19.15480698340495358360057327740, −17.759853877318785966220887025366, −15.52774644829846216199192434779, −14.95302825195012228590214442695, −13.76268107671948964676107906301, −12.93915836465682557160010546979, −11.77114007490005612940103780076, −10.16918249206535294941175390578, −9.36950217311456810123341655819, −7.13439628149696372076140982136, −6.53241529812561843015462419478, −4.46503437095183538920807141189, −2.96083412827119223571645419181, −2.25917748017797984374966874233, 2.58550026582726703548981240784, 3.86239846100532881359260545404, 5.04321601169484943665681990121, 6.51334204731557476831486469508, 8.1142414923373455551394058076, 8.95330134744128491634141762911, 10.46884315368397638365300949669, 12.476143848727603518695585444962, 13.15189756742455887049026674375, 14.1720108559689836851560041264, 15.44356005857221297516101010296, 16.268365111222899396322197135084, 17.145952255963093126091226593819, 19.29038468870665925822917521792, 20.1321236823662080258950168998, 21.26265582820026060412208409368, 21.9801846060404336083106026651, 23.34479467016783914228369027481, 24.63866094704444012232236413789, 25.07585750639897588169466912035, 26.31248606014543580972442702755, 27.102976920463142479744946872334, 28.832369977495944472543776928667, 29.75812719585743694116714496555, 31.107000840400357174581466544414

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