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Properties

Label 1-89-89.30-r1-0-0
Degree $1$
Conductor $89$
Sign $0.621 - 0.783i$
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  + (0.415 − 0.909i)2-s + (0.997 − 0.0713i)3-s + (−0.654 − 0.755i)4-s + (0.281 + 0.959i)5-s + (0.349 − 0.936i)6-s + (0.877 + 0.479i)7-s + (−0.959 + 0.281i)8-s + (0.989 − 0.142i)9-s + (0.989 + 0.142i)10-s + (0.959 + 0.281i)11-s + (−0.707 − 0.707i)12-s + (−0.0713 − 0.997i)13-s + (0.800 − 0.599i)14-s + (0.349 + 0.936i)15-s + (−0.142 + 0.989i)16-s + (0.909 − 0.415i)17-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (0.997 − 0.0713i)3-s + (−0.654 − 0.755i)4-s + (0.281 + 0.959i)5-s + (0.349 − 0.936i)6-s + (0.877 + 0.479i)7-s + (−0.959 + 0.281i)8-s + (0.989 − 0.142i)9-s + (0.989 + 0.142i)10-s + (0.959 + 0.281i)11-s + (−0.707 − 0.707i)12-s + (−0.0713 − 0.997i)13-s + (0.800 − 0.599i)14-s + (0.349 + 0.936i)15-s + (−0.142 + 0.989i)16-s + (0.909 − 0.415i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.621 - 0.783i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ 0.621 - 0.783i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.799120397 - 1.353450102i\)
\(L(\frac12)\) \(\approx\) \(2.799120397 - 1.353450102i\)
\(L(1)\) \(\approx\) \(1.823690997 - 0.6976228302i\)
\(L(1)\) \(\approx\) \(1.823690997 - 0.6976228302i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (0.415 - 0.909i)T \)
3 \( 1 + (0.997 - 0.0713i)T \)
5 \( 1 + (0.281 + 0.959i)T \)
7 \( 1 + (0.877 + 0.479i)T \)
11 \( 1 + (0.959 + 0.281i)T \)
13 \( 1 + (-0.0713 - 0.997i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (-0.800 - 0.599i)T \)
23 \( 1 + (-0.599 + 0.800i)T \)
29 \( 1 + (-0.479 + 0.877i)T \)
31 \( 1 + (-0.599 - 0.800i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.0713 - 0.997i)T \)
43 \( 1 + (-0.479 - 0.877i)T \)
47 \( 1 + (-0.755 + 0.654i)T \)
53 \( 1 + (0.755 + 0.654i)T \)
59 \( 1 + (-0.997 - 0.0713i)T \)
61 \( 1 + (0.212 + 0.977i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (-0.281 + 0.959i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (-0.989 - 0.142i)T \)
83 \( 1 + (-0.349 + 0.936i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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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

−30.659317229480405167794854854490, −29.821296842415312788918482774957, −27.88692648268130021194832485360, −27.05458943491895082985809484282, −26.01023632595302835236355111197, −24.93753871235874314640313672467, −24.339712487373544036561605477565, −23.41833554862753051168681470487, −21.5854419130257964377675996742, −21.091472643898666695273669671330, −19.86732115998783807780832196200, −18.458862151436771996386683551130, −16.9511848696801176776136022840, −16.429802836586742957431947562105, −14.74241930463461194329414961285, −14.23119143294935351730112219485, −13.161756664327914962660800270406, −11.95293500625842861403271004066, −9.748513196317116978293273252676, −8.61743892143487162415328406033, −7.90801350257525786610153544826, −6.36954770719195756067244758432, −4.67104434813384713171155399014, −3.86955247779711039946645801626, −1.59765892221084299205473063221, 1.653979743445574391303438040770, 2.77407833635514674851114251204, 3.9784814529959297108658756032, 5.658670066657858478207222648177, 7.43599876199172080907250400230, 8.92232227541743201544279823753, 9.99352035921359276920820710660, 11.177307426542708034674724819, 12.429570764267552956804659988864, 13.73755826279856334124943356014, 14.66083372372391459821265770453, 15.14904315841449214205187656222, 17.68162381122291976872831325387, 18.50852910826330743090596231471, 19.518725054628212715186655942180, 20.49678303398487274274805730861, 21.545102515523402675465025655174, 22.28699070467483915870385820533, 23.66472649758971736128372517489, 24.91135282383587704228229620947, 25.85150654487847647718385627775, 27.36814617746572195965458709487, 27.69991692063884845304000936104, 29.71384181089070678533687715571, 30.0987263342886496804052115982

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