Properties

Degree 1
Conductor 89
Sign $0.928 + 0.371i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $0.928 + 0.371i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (37, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 89,\ (1:\ ),\ 0.928 + 0.371i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.973475381 + 0.7643857403i$
$L(\frac12,\chi)$  $\approx$  $3.973475381 + 0.7643857403i$
$L(\chi,1)$  $\approx$  2.465620914 + 0.3294097200i
$L(1,\chi)$  $\approx$  2.465620914 + 0.3294097200i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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