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Properties

Label 1-79-79.53-r1-0-0
Degree $1$
Conductor $79$
Sign $0.998 - 0.0565i$
Analytic cond. $8.48972$
Root an. cond. $8.48972$
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  + (0.948 − 0.316i)2-s + (0.996 − 0.0804i)3-s + (0.799 − 0.600i)4-s + (0.278 + 0.960i)5-s + (0.919 − 0.391i)6-s + (−0.428 + 0.903i)7-s + (0.568 − 0.822i)8-s + (0.987 − 0.160i)9-s + (0.568 + 0.822i)10-s + (0.692 + 0.721i)11-s + (0.748 − 0.663i)12-s + (−0.919 − 0.391i)13-s + (−0.120 + 0.992i)14-s + (0.354 + 0.935i)15-s + (0.278 − 0.960i)16-s + (−0.120 − 0.992i)17-s + ⋯
L(s)  = 1  + (0.948 − 0.316i)2-s + (0.996 − 0.0804i)3-s + (0.799 − 0.600i)4-s + (0.278 + 0.960i)5-s + (0.919 − 0.391i)6-s + (−0.428 + 0.903i)7-s + (0.568 − 0.822i)8-s + (0.987 − 0.160i)9-s + (0.568 + 0.822i)10-s + (0.692 + 0.721i)11-s + (0.748 − 0.663i)12-s + (−0.919 − 0.391i)13-s + (−0.120 + 0.992i)14-s + (0.354 + 0.935i)15-s + (0.278 − 0.960i)16-s + (−0.120 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.998 - 0.0565i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ 0.998 - 0.0565i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.981488930 - 0.1127215123i\)
\(L(\frac12)\) \(\approx\) \(3.981488930 - 0.1127215123i\)
\(L(1)\) \(\approx\) \(2.495379437 - 0.1279703292i\)
\(L(1)\) \(\approx\) \(2.495379437 - 0.1279703292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.948 - 0.316i)T \)
3 \( 1 + (0.996 - 0.0804i)T \)
5 \( 1 + (0.278 + 0.960i)T \)
7 \( 1 + (-0.428 + 0.903i)T \)
11 \( 1 + (0.692 + 0.721i)T \)
13 \( 1 + (-0.919 - 0.391i)T \)
17 \( 1 + (-0.120 - 0.992i)T \)
19 \( 1 + (-0.845 - 0.534i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.632 - 0.774i)T \)
31 \( 1 + (-0.200 + 0.979i)T \)
37 \( 1 + (0.0402 - 0.999i)T \)
41 \( 1 + (0.970 + 0.239i)T \)
43 \( 1 + (-0.692 + 0.721i)T \)
47 \( 1 + (0.0402 + 0.999i)T \)
53 \( 1 + (0.996 + 0.0804i)T \)
59 \( 1 + (-0.799 - 0.600i)T \)
61 \( 1 + (-0.885 + 0.464i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (-0.568 + 0.822i)T \)
73 \( 1 + (-0.919 + 0.391i)T \)
83 \( 1 + (0.799 - 0.600i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (0.885 - 0.464i)T \)
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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

−31.19810605857676645221879743837, −29.85382003294739447687195213092, −29.31447360042910112185981338458, −27.46397948658630723281659590485, −26.24479404437068864221510001088, −25.37044523468246805451558167369, −24.326935386071937580781511072680, −23.68197591253712166020996072599, −21.96317881328653087327906968626, −21.22407302916777874114898438839, −20.02233673586323147737213883196, −19.475412010394310524613908039575, −17.0604896099547262415179252803, −16.44916481158917333255930249089, −15.05244354620252478059607102291, −13.95603366753778951259875768757, −13.225068922629716485554270564347, −12.14578640691844733650988825156, −10.27328236555705716873502166212, −8.828145656877882635852938153325, −7.637236827933848018607246025, −6.23830163421459634823575896394, −4.49098431525697461210130713234, −3.596389346409099070296424332605, −1.771627309976480566646381271320, 2.23933403024223979640157070862, 2.907638380526986143336438014857, 4.50510319126903745202170771668, 6.31465895881721791128850773416, 7.31258049336690335058498614136, 9.30331001768519665641411624227, 10.29689741281446110834422431235, 11.95012534850251695696008444317, 12.93976812435371181360903269670, 14.29120041465833185862743963087, 14.852967932515795301134989049442, 15.836337878031411002183841857545, 17.99012570716029076392075039066, 19.21886724264275295129066383370, 19.89806117855751889473094716929, 21.286861428655952380567090882368, 22.10566516193993192204228813250, 22.99226831733807102368124385709, 24.759840014671910298998820080, 25.13703508447381180533246587533, 26.31124593957576377151927776603, 27.7257704396535464251913037731, 29.166365992555334269212787140548, 30.12509302125876323314967027232, 30.82196548913587510345557011164

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