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Properties

Label 1-89-89.77-r1-0-0
Degree $1$
Conductor $89$
Sign $0.928 - 0.371i$
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  + 2-s + (0.707 − 0.707i)3-s + 4-s + i·5-s + (0.707 − 0.707i)6-s + (0.707 − 0.707i)7-s + 8-s − i·9-s + i·10-s − 11-s + (0.707 − 0.707i)12-s + (0.707 + 0.707i)13-s + (0.707 − 0.707i)14-s + (0.707 + 0.707i)15-s + 16-s + i·17-s + ⋯
L(s)  = 1  + 2-s + (0.707 − 0.707i)3-s + 4-s + i·5-s + (0.707 − 0.707i)6-s + (0.707 − 0.707i)7-s + 8-s − i·9-s + i·10-s − 11-s + (0.707 − 0.707i)12-s + (0.707 + 0.707i)13-s + (0.707 − 0.707i)14-s + (0.707 + 0.707i)15-s + 16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.928 - 0.371i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ 0.928 - 0.371i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.973475381 - 0.7643857403i\)
\(L(\frac12)\) \(\approx\) \(3.973475381 - 0.7643857403i\)
\(L(1)\) \(\approx\) \(2.465620914 - 0.3294097200i\)
\(L(1)\) \(\approx\) \(2.465620914 - 0.3294097200i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + iT \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 - T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + iT \)
53 \( 1 + iT \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 + T \)
71 \( 1 + iT \)
73 \( 1 - T \)
79 \( 1 - iT \)
83 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + T \)
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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

−30.82467698958986209618739820664, −29.385307184303985020029164512211, −28.177066022595818422347175271882, −27.38620805175636966865747349231, −25.67303627433568470667470398568, −25.03900901305365354384667391917, −24.048950107940709341462976954171, −22.84875860055171351092193156470, −21.48332015044776631805859201507, −20.82330726821278965276989995075, −20.282620643699782422740312610438, −18.71661061741529810007129205857, −16.88731409119533152430310804981, −15.69501537291678895672074707247, −15.19422390616372599607161249038, −13.7749679502098855619650860448, −12.93022555209322854841088806713, −11.58906581822982212040973454423, −10.31454898265347313059468260591, −8.71008784519031654058323109003, −7.78240643526686703584472969993, −5.47542664634010117305288038718, −4.88480945728969854803306038230, −3.389940091642007092438843167217, −1.94947107553935518895804654054, 1.77214373107954696908489956934, 3.01675818220542927261427559108, 4.32110965131381855137595787689, 6.2500561670825149395352038488, 7.18596910276973867327237035687, 8.25678365884363584747030603544, 10.4827415955828752512521207474, 11.347488304717666620620637655794, 12.90110394917559827907835657734, 13.75141224011110188506024314280, 14.62207279975742729915226098791, 15.48239057320616231356721171765, 17.26025743610620635808768207717, 18.588072261729419338161863732800, 19.5445309974520568185770659450, 20.81283771158204284915296225666, 21.48660097759902934627702162277, 23.21052858176467217987519619497, 23.59642152322914028610397078089, 24.71527679177104034203712882486, 26.07986805306637874349842549724, 26.39928905719738365112376937224, 28.5000389807223331340029126213, 29.67967882573099746355673967656, 30.5214691733561436117717758355

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