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Properties

Label 1-79-79.48-r1-0-0
Degree $1$
Conductor $79$
Sign $-0.999 + 0.0242i$
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  + (0.692 − 0.721i)2-s + (0.200 + 0.979i)3-s + (−0.0402 − 0.999i)4-s + (−0.996 − 0.0804i)5-s + (0.845 + 0.534i)6-s + (−0.948 − 0.316i)7-s + (−0.748 − 0.663i)8-s + (−0.919 + 0.391i)9-s + (−0.748 + 0.663i)10-s + (0.428 − 0.903i)11-s + (0.970 − 0.239i)12-s + (−0.845 + 0.534i)13-s + (−0.885 + 0.464i)14-s + (−0.120 − 0.992i)15-s + (−0.996 + 0.0804i)16-s + (−0.885 − 0.464i)17-s + ⋯
L(s)  = 1  + (0.692 − 0.721i)2-s + (0.200 + 0.979i)3-s + (−0.0402 − 0.999i)4-s + (−0.996 − 0.0804i)5-s + (0.845 + 0.534i)6-s + (−0.948 − 0.316i)7-s + (−0.748 − 0.663i)8-s + (−0.919 + 0.391i)9-s + (−0.748 + 0.663i)10-s + (0.428 − 0.903i)11-s + (0.970 − 0.239i)12-s + (−0.845 + 0.534i)13-s + (−0.885 + 0.464i)14-s + (−0.120 − 0.992i)15-s + (−0.996 + 0.0804i)16-s + (−0.885 − 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.999 + 0.0242i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ -0.999 + 0.0242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006447027667 - 0.5318761813i\)
\(L(\frac12)\) \(\approx\) \(0.006447027667 - 0.5318761813i\)
\(L(1)\) \(\approx\) \(0.8446079427 - 0.3123326795i\)
\(L(1)\) \(\approx\) \(0.8446079427 - 0.3123326795i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.692 - 0.721i)T \)
3 \( 1 + (0.200 + 0.979i)T \)
5 \( 1 + (-0.996 - 0.0804i)T \)
7 \( 1 + (-0.948 - 0.316i)T \)
11 \( 1 + (0.428 - 0.903i)T \)
13 \( 1 + (-0.845 + 0.534i)T \)
17 \( 1 + (-0.885 - 0.464i)T \)
19 \( 1 + (0.987 - 0.160i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.799 + 0.600i)T \)
31 \( 1 + (0.278 + 0.960i)T \)
37 \( 1 + (0.632 + 0.774i)T \)
41 \( 1 + (-0.568 + 0.822i)T \)
43 \( 1 + (-0.428 - 0.903i)T \)
47 \( 1 + (0.632 - 0.774i)T \)
53 \( 1 + (0.200 - 0.979i)T \)
59 \( 1 + (0.0402 - 0.999i)T \)
61 \( 1 + (0.354 - 0.935i)T \)
67 \( 1 + (-0.970 + 0.239i)T \)
71 \( 1 + (0.748 + 0.663i)T \)
73 \( 1 + (-0.845 - 0.534i)T \)
83 \( 1 + (-0.0402 - 0.999i)T \)
89 \( 1 + (-0.748 + 0.663i)T \)
97 \( 1 + (-0.354 + 0.935i)T \)
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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

−31.45901980448926191835146795323, −30.58284149779454940591276913875, −29.66238571151673678768861069083, −28.30696186405034491315367767953, −26.73688795433598838240569969575, −25.770594999592457367869574579919, −24.77193200442154691075972262091, −23.960535125214628289578729464144, −22.76777590421888737600900619609, −22.33314919762657109272958444232, −20.241353860260273663663459416819, −19.51081913788623857860729065089, −18.1157701536716745434593671611, −16.98047003677783212574355959803, −15.57263348496333606807022939163, −14.82754929856859338728808869605, −13.37519840008384972303248636993, −12.433565960860932743668526462893, −11.70450266708239046133658822401, −9.27188358574006486205506845381, −7.77532447249369345041407874385, −7.09611238952842793891777766370, −5.82090166175133151994768420486, −4.00646156916714745997271635827, −2.65293343294108422270189723339, 0.18911767349457324800814709818, 2.95296878553193828042970762942, 3.87071720121762764393020854212, 4.99723209166658932215478075316, 6.73340475434189864735434078739, 8.81748646339601632953366379426, 9.905128198436811500117764212235, 11.135871834999962443261774541028, 12.0289059752973022860561171419, 13.557247049093066933807806613709, 14.62875976860092160161926448723, 15.809812844594739613474744759925, 16.547258280294438725127335031409, 18.8086132703828578873115720810, 19.85006060279354551361676525940, 20.29986591467310693065688394749, 21.99423275859988409953845720182, 22.31446871745165077650417842306, 23.57410548699702838917300373793, 24.70883181939098347107563283080, 26.60613252994021930109748262791, 27.03685616977295110101005163652, 28.38962565254230244207618966944, 29.16443503427942794654645150613, 30.560609762907683892583673554032

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