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Properties

Label 1-89-89.10-r0-0-0
Degree $1$
Conductor $89$
Sign $0.987 - 0.155i$
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  + (−0.654 − 0.755i)2-s + (0.989 − 0.142i)3-s + (−0.142 + 0.989i)4-s + (−0.841 + 0.540i)5-s + (−0.755 − 0.654i)6-s + (0.540 + 0.841i)7-s + (0.841 − 0.540i)8-s + (0.959 − 0.281i)9-s + (0.959 + 0.281i)10-s + (0.841 + 0.540i)11-s + i·12-s + (−0.989 + 0.142i)13-s + (0.281 − 0.959i)14-s + (−0.755 + 0.654i)15-s + (−0.959 − 0.281i)16-s + (0.654 − 0.755i)17-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (0.989 − 0.142i)3-s + (−0.142 + 0.989i)4-s + (−0.841 + 0.540i)5-s + (−0.755 − 0.654i)6-s + (0.540 + 0.841i)7-s + (0.841 − 0.540i)8-s + (0.959 − 0.281i)9-s + (0.959 + 0.281i)10-s + (0.841 + 0.540i)11-s + i·12-s + (−0.989 + 0.142i)13-s + (0.281 − 0.959i)14-s + (−0.755 + 0.654i)15-s + (−0.959 − 0.281i)16-s + (0.654 − 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ 0.987 - 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9127087648 - 0.07162054058i\)
\(L(\frac12)\) \(\approx\) \(0.9127087648 - 0.07162054058i\)
\(L(1)\) \(\approx\) \(0.9496315737 - 0.1271760229i\)
\(L(1)\) \(\approx\) \(0.9496315737 - 0.1271760229i\)

Euler product

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

−30.676507746147261037969149983011, −29.55946642230561986994986922462, −27.80115552631743810820905942473, −27.28170332457990313297153080128, −26.46239585826866675012131598806, −25.36796753573971157836819366412, −24.07040558102105320273809488843, −23.93383704760466006896025375885, −22.073590225185663314131787875760, −20.40449507138095179648039190350, −19.74026644347661864773615919418, −18.9722124667884560639153856590, −17.309198625812646717248053607476, −16.47909877148079080530763745254, −15.22333383010100597280442438855, −14.47122596320990383471350691663, −13.307934733860501642529988062607, −11.47256811978672720158455179915, −10.0238406544955436391693992804, −8.87937833743713476208905308537, −7.89545671923952756718166423128, −7.117604857218311911814188995878, −4.98929002966046638085251714469, −3.75235195033865535728398504102, −1.36009656007219707076576548506, 1.910155509259043959130979878682, 3.10015659341734588941507358957, 4.38233389378792073564487697350, 7.12391638177435799641002720448, 7.99231155174228463983867981314, 9.104157795584194038676625677285, 10.18806863001991452582817822771, 11.86286742428166684274625874078, 12.31240634224028928521844116713, 14.213341746337831010910406173158, 15.01793620565759581710171530209, 16.43766682916441944822847582357, 18.08335523684397471143521323157, 18.81152532768674833538995741428, 19.70084543550041067824495260623, 20.58156470370328652801570134371, 21.72802496600993659883468938692, 22.788888600066131723823399479789, 24.61190467535455447981073339289, 25.30768098146715488960739487574, 26.686013328322366584392415873320, 27.184604914450538450970899552181, 28.18634648048248200102045074288, 29.72450862016584650050090072184, 30.48462537618216024377192648938

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