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Properties

Label 1-79-79.32-r0-0-0
Degree $1$
Conductor $79$
Sign $0.728 + 0.685i$
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  + (0.987 + 0.160i)2-s + (−0.0402 + 0.999i)3-s + (0.948 + 0.316i)4-s + (0.799 − 0.600i)5-s + (−0.200 + 0.979i)6-s + (−0.845 − 0.534i)7-s + (0.885 + 0.464i)8-s + (−0.996 − 0.0804i)9-s + (0.885 − 0.464i)10-s + (−0.919 + 0.391i)11-s + (−0.354 + 0.935i)12-s + (−0.200 − 0.979i)13-s + (−0.748 − 0.663i)14-s + (0.568 + 0.822i)15-s + (0.799 + 0.600i)16-s + (−0.748 + 0.663i)17-s + ⋯
L(s)  = 1  + (0.987 + 0.160i)2-s + (−0.0402 + 0.999i)3-s + (0.948 + 0.316i)4-s + (0.799 − 0.600i)5-s + (−0.200 + 0.979i)6-s + (−0.845 − 0.534i)7-s + (0.885 + 0.464i)8-s + (−0.996 − 0.0804i)9-s + (0.885 − 0.464i)10-s + (−0.919 + 0.391i)11-s + (−0.354 + 0.935i)12-s + (−0.200 − 0.979i)13-s + (−0.748 − 0.663i)14-s + (0.568 + 0.822i)15-s + (0.799 + 0.600i)16-s + (−0.748 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.728 + 0.685i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.728 + 0.685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.509615771 + 0.5985732682i\)
\(L(\frac12)\) \(\approx\) \(1.509615771 + 0.5985732682i\)
\(L(1)\) \(\approx\) \(1.594255522 + 0.4551550741i\)
\(L(1)\) \(\approx\) \(1.594255522 + 0.4551550741i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.987 + 0.160i)T \)
3 \( 1 + (-0.0402 + 0.999i)T \)
5 \( 1 + (0.799 - 0.600i)T \)
7 \( 1 + (-0.845 - 0.534i)T \)
11 \( 1 + (-0.919 + 0.391i)T \)
13 \( 1 + (-0.200 - 0.979i)T \)
17 \( 1 + (-0.748 + 0.663i)T \)
19 \( 1 + (0.278 + 0.960i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.428 - 0.903i)T \)
31 \( 1 + (-0.632 + 0.774i)T \)
37 \( 1 + (0.692 - 0.721i)T \)
41 \( 1 + (0.120 + 0.992i)T \)
43 \( 1 + (-0.919 - 0.391i)T \)
47 \( 1 + (0.692 + 0.721i)T \)
53 \( 1 + (-0.0402 - 0.999i)T \)
59 \( 1 + (0.948 - 0.316i)T \)
61 \( 1 + (-0.970 - 0.239i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (0.885 + 0.464i)T \)
73 \( 1 + (-0.200 + 0.979i)T \)
83 \( 1 + (0.948 + 0.316i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (-0.970 - 0.239i)T \)
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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

−31.1154722647625589897613479810, −29.88025249261197637092348290240, −29.07288723477221238908369584505, −28.64289376092123456637153365261, −26.15030115605559284906407936066, −25.50011961777440602412965242953, −24.36917807973574748996856282238, −23.494824841049802197393180952046, −22.25827079319057039062199604581, −21.63949111332418072980542142079, −20.0889868151165685535773892548, −18.98009054128769986496112201423, −18.093113662807821158371991924591, −16.4829037415332385627282503221, −15.179378805411324543450648262530, −13.69817461288517851821892065598, −13.37102156995461048987818237276, −12.02342475502995727246413714733, −10.92549964534950251314613706768, −9.32119182882665181442706659718, −7.23391500265957101290996887717, −6.37170188322420997497842772424, −5.32282387884200089259326173100, −3.007638702222971672832359524775, −2.137002933704508080192393930202, 2.59803211719173761263835674447, 4.04369811024072534402675296558, 5.24639204678134643715743932920, 6.24993402579303077380396203179, 8.13945052196111987118588150784, 9.92660775235331384130224569922, 10.612555692267543437072184564314, 12.47937729131483785196663554983, 13.3029216010856441646393988356, 14.54882951264470435846256008331, 15.77019049539882185159614322050, 16.51748485584680603634494822243, 17.61445690569536137980468991550, 19.96929526117736133570200177807, 20.55472085473850142267336638943, 21.60519079510753026982600716403, 22.53088989382095648801817599751, 23.42513075747556099033084222486, 24.90499230668154210256853006359, 25.764498229711042961909943931128, 26.72002041318606374984026782954, 28.55021965650963766885047024276, 28.94991078092412853329585290492, 30.29606818027948289939293836534, 31.73021425272780754421931228654

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