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Properties

Label 1-79-79.26-r0-0-0
Degree $1$
Conductor $79$
Sign $0.944 - 0.328i$
Analytic cond. $0.366874$
Root an. cond. $0.366874$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.944 - 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.310862636 - 0.2217081046i\)
\(L(\frac12)\) \(\approx\) \(1.310862636 - 0.2217081046i\)
\(L(1)\) \(\approx\) \(1.369294340 - 0.1766153088i\)
\(L(1)\) \(\approx\) \(1.369294340 - 0.1766153088i\)

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

−31.59948715973941998966559532498, −29.92909358056389036481738498539, −29.29067095019980255312827938850, −28.170713000538587999386343403844, −27.148172750062714448790001697842, −25.218210257138906223517619816606, −24.48399688614748994193588829246, −23.83368987611298911828282346721, −22.44103297854839021931987750596, −21.65729727736459325118222497626, −20.838717679754821093086433842001, −19.177352571063123390501936518199, −17.53439138469023810127265320956, −16.7103165218087269669233642290, −15.80701251047149217210792788220, −14.40929787847206161526424075125, −13.06040170232532005097821803769, −12.02549253102257288249986800552, −11.42385872445042117917091694186, −9.4172359593038585442806606258, −7.79236130944077129365190831128, −6.15971627053376947116629159229, −5.35978074281895904602274954774, −4.2719917877821442588440375374, −1.94909690305608978653562166493, 1.84544500757720920501603815431, 3.82729927941831135229941462618, 4.97859555783026935438727815350, 6.47450258386241233798362931051, 7.195138109379113301155990144809, 10.17487561730313384556641812017, 10.64459928616803950818563990737, 11.87779709181975138050538759144, 12.96784910615332748966884932543, 14.42584331110439679102480691038, 15.14063167822986140536333661548, 16.81996758927689319083540283118, 17.68288029151621084467866789466, 19.18155493057713262087893193119, 20.39203951856967294339728196065, 21.792544238656226206953264804445, 22.316020178834918388205198305977, 23.351118332790141855076212743662, 24.13738727602897240977832501984, 25.57200043175608655042423429412, 27.018437755260570933301962581890, 28.08064201520689899821872585957, 29.313335122722049802044484905448, 30.11780685430721190780361168644, 30.58885349252142180426186951895

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