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Properties

Label 1-89-89.27-r1-0-0
Degree $1$
Conductor $89$
Sign $0.961 - 0.276i$
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  + (−0.959 − 0.281i)2-s + (0.977 + 0.212i)3-s + (0.841 + 0.540i)4-s + (−0.755 + 0.654i)5-s + (−0.877 − 0.479i)6-s + (0.0713 − 0.997i)7-s + (−0.654 − 0.755i)8-s + (0.909 + 0.415i)9-s + (0.909 − 0.415i)10-s + (0.654 − 0.755i)11-s + (0.707 + 0.707i)12-s + (0.212 − 0.977i)13-s + (−0.349 + 0.936i)14-s + (−0.877 + 0.479i)15-s + (0.415 + 0.909i)16-s + (0.281 + 0.959i)17-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.977 + 0.212i)3-s + (0.841 + 0.540i)4-s + (−0.755 + 0.654i)5-s + (−0.877 − 0.479i)6-s + (0.0713 − 0.997i)7-s + (−0.654 − 0.755i)8-s + (0.909 + 0.415i)9-s + (0.909 − 0.415i)10-s + (0.654 − 0.755i)11-s + (0.707 + 0.707i)12-s + (0.212 − 0.977i)13-s + (−0.349 + 0.936i)14-s + (−0.877 + 0.479i)15-s + (0.415 + 0.909i)16-s + (0.281 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.961 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.961 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.961 - 0.276i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ 0.961 - 0.276i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.463274227 - 0.2063215591i\)
\(L(\frac12)\) \(\approx\) \(1.463274227 - 0.2063215591i\)
\(L(1)\) \(\approx\) \(1.006958888 - 0.07534174884i\)
\(L(1)\) \(\approx\) \(1.006958888 - 0.07534174884i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.959 - 0.281i)T \)
3 \( 1 + (0.977 + 0.212i)T \)
5 \( 1 + (-0.755 + 0.654i)T \)
7 \( 1 + (0.0713 - 0.997i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (0.212 - 0.977i)T \)
17 \( 1 + (0.281 + 0.959i)T \)
19 \( 1 + (0.349 + 0.936i)T \)
23 \( 1 + (0.936 - 0.349i)T \)
29 \( 1 + (0.997 + 0.0713i)T \)
31 \( 1 + (0.936 + 0.349i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.212 - 0.977i)T \)
43 \( 1 + (0.997 - 0.0713i)T \)
47 \( 1 + (0.540 - 0.841i)T \)
53 \( 1 + (-0.540 - 0.841i)T \)
59 \( 1 + (-0.977 + 0.212i)T \)
61 \( 1 + (-0.599 + 0.800i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.755 + 0.654i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (-0.909 + 0.415i)T \)
83 \( 1 + (0.877 + 0.479i)T \)
97 \( 1 + (-0.654 - 0.755i)T \)
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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

−30.468133639869359769652914956417, −28.95187899430909142292688502063, −28.00060391909207343833873542360, −27.16641652661064501617555701391, −26.08736532208530560479022039182, −24.98323774628302063453193957520, −24.53588887899364793600771664487, −23.28733469393395054920944698135, −21.3155637023914676400509438415, −20.35497771969129654626981963744, −19.405409956430313467687696504460, −18.72329948421491645975808644499, −17.44040143396958762632558782568, −15.938078127296062496331932510203, −15.35413400041364286377744763758, −14.14263805722736814838719779469, −12.36716865892044770118881828092, −11.495366134292542953550984244135, −9.35933708928821086101307688112, −9.04258229299447287247996890424, −7.78866455886759664931158549057, −6.73640656489362559323222281684, −4.71833037299613541454609878012, −2.75490425500137803117584890742, −1.25527687420074306229806838466, 1.07901156677222961731904334046, 3.08505011207761590279860184290, 3.84121165319741181818263934026, 6.68677206691838731983177152763, 7.83578155897151731194428023640, 8.54866112328874516746883186811, 10.206570879245265422859468773415, 10.77541687072907317454058483049, 12.32590632564934720443584106801, 13.89536594538516200103404338191, 15.05554555827140801433981767694, 16.09249692301899938012476958640, 17.27909998193359050328742468601, 18.76157268576386239867917454243, 19.43937530356600619961353855871, 20.267492322199271317457710509691, 21.27088619643389504425888892902, 22.68838732457186282719643881796, 24.18249892827262309703240002807, 25.29795534811825894605101597675, 26.30157025000275911155941706927, 27.10318747888169021911852923369, 27.52910599929166175109135059995, 29.35187845477251717554709818433, 30.30036554448486440434244990889

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