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Properties

Label 1-179-179.67-r0-0-0
Degree $1$
Conductor $179$
Sign $0.292 + 0.956i$
Analytic cond. $0.831272$
Root an. cond. $0.831272$
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  + (−0.994 + 0.105i)2-s + (0.427 + 0.904i)3-s + (0.977 − 0.210i)4-s + (−0.458 − 0.888i)5-s + (−0.520 − 0.854i)6-s + (−0.737 − 0.675i)7-s + (−0.949 + 0.312i)8-s + (−0.635 + 0.772i)9-s + (0.550 + 0.835i)10-s + (0.0176 + 0.999i)11-s + (0.607 + 0.794i)12-s + (0.880 + 0.474i)13-s + (0.804 + 0.593i)14-s + (0.607 − 0.794i)15-s + (0.911 − 0.411i)16-s + (0.295 + 0.955i)17-s + ⋯
L(s)  = 1  + (−0.994 + 0.105i)2-s + (0.427 + 0.904i)3-s + (0.977 − 0.210i)4-s + (−0.458 − 0.888i)5-s + (−0.520 − 0.854i)6-s + (−0.737 − 0.675i)7-s + (−0.949 + 0.312i)8-s + (−0.635 + 0.772i)9-s + (0.550 + 0.835i)10-s + (0.0176 + 0.999i)11-s + (0.607 + 0.794i)12-s + (0.880 + 0.474i)13-s + (0.804 + 0.593i)14-s + (0.607 − 0.794i)15-s + (0.911 − 0.411i)16-s + (0.295 + 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5681868004 + 0.4202325751i\)
\(L(\frac12)\) \(\approx\) \(0.5681868004 + 0.4202325751i\)
\(L(1)\) \(\approx\) \(0.6779528764 + 0.2261304741i\)
\(L(1)\) \(\approx\) \(0.6779528764 + 0.2261304741i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.994 + 0.105i)T \)
3 \( 1 + (0.427 + 0.904i)T \)
5 \( 1 + (-0.458 - 0.888i)T \)
7 \( 1 + (-0.737 - 0.675i)T \)
11 \( 1 + (0.0176 + 0.999i)T \)
13 \( 1 + (0.880 + 0.474i)T \)
17 \( 1 + (0.295 + 0.955i)T \)
19 \( 1 + (0.844 + 0.535i)T \)
23 \( 1 + (0.158 + 0.987i)T \)
29 \( 1 + (0.997 + 0.0705i)T \)
31 \( 1 + (0.960 - 0.278i)T \)
37 \( 1 + (0.997 - 0.0705i)T \)
41 \( 1 + (0.760 - 0.648i)T \)
43 \( 1 + (-0.579 - 0.815i)T \)
47 \( 1 + (-0.783 + 0.621i)T \)
53 \( 1 + (-0.969 + 0.244i)T \)
59 \( 1 + (-0.329 + 0.944i)T \)
61 \( 1 + (0.227 - 0.973i)T \)
67 \( 1 + (-0.949 - 0.312i)T \)
71 \( 1 + (-0.999 - 0.0352i)T \)
73 \( 1 + (0.938 - 0.345i)T \)
79 \( 1 + (-0.394 + 0.918i)T \)
83 \( 1 + (-0.825 - 0.564i)T \)
89 \( 1 + (-0.994 - 0.105i)T \)
97 \( 1 + (-0.688 + 0.725i)T \)
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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

−26.83536275203246639659867590575, −26.33483986268049758134977040789, −25.25317385725155848044097914613, −24.741658099948436691685684672790, −23.45161715820706847180299826306, −22.43235054148712101084291585865, −21.1263793716726553991829654143, −19.9041151150417054096666150828, −19.228799125064142715817958270154, −18.38864910809405639019845234843, −18.02944930361192304395388642304, −16.30884749515113785013318233268, −15.59219951113349046563568436332, −14.37888726504936966403800544946, −13.16391294557881153483749655033, −11.89982974448525017228799801788, −11.22881882488948089593202380374, −9.85908411721453573101882870007, −8.68366868549789897960327221314, −7.91248658186908020898229953665, −6.7088023308453546641633699972, −6.07906884802791720666073523400, −3.05001995698623262472747786712, −2.86724653578341332716924838203, −0.823688650471138039865422896040, 1.43520422560118741672597764916, 3.3052922946095545450913442013, 4.38420394896392272465842680310, 5.94310324540046740891893553820, 7.46242774460657282617190818297, 8.376897017802010359768307313661, 9.45174512214889894127628143513, 10.05525325710903472914209713325, 11.24302287903725996619120545763, 12.44244691423598296653355534960, 13.83157784060194980187740788396, 15.24412283370059692645798634489, 15.95550224877854197797909371559, 16.65986672423718681949169784744, 17.53247629233100780483586302137, 19.13604748208551094507669869300, 19.829782090940247384224491091292, 20.550818857444781770876584743926, 21.3012942695750516192697248452, 22.92260206323728029847470408894, 23.76106846445856854614192387100, 25.16047608181609014227057137824, 25.75999855805486885744205927494, 26.639957087957136645435535987883, 27.48854390268159390875951050395

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