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Properties

Label 1-101-101.2-r1-0-0
Degree $1$
Conductor $101$
Sign $0.970 + 0.241i$
Analytic cond. $10.8539$
Root an. cond. $10.8539$
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  + (0.998 + 0.0627i)2-s + (−0.368 − 0.929i)3-s + (0.992 + 0.125i)4-s + (0.0627 + 0.998i)5-s + (−0.309 − 0.951i)6-s + (0.844 + 0.535i)7-s + (0.982 + 0.187i)8-s + (−0.728 + 0.684i)9-s + i·10-s + (0.684 + 0.728i)11-s + (−0.248 − 0.968i)12-s + (−0.535 − 0.844i)13-s + (0.809 + 0.587i)14-s + (0.904 − 0.425i)15-s + (0.968 + 0.248i)16-s + (−0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (0.998 + 0.0627i)2-s + (−0.368 − 0.929i)3-s + (0.992 + 0.125i)4-s + (0.0627 + 0.998i)5-s + (−0.309 − 0.951i)6-s + (0.844 + 0.535i)7-s + (0.982 + 0.187i)8-s + (−0.728 + 0.684i)9-s + i·10-s + (0.684 + 0.728i)11-s + (−0.248 − 0.968i)12-s + (−0.535 − 0.844i)13-s + (0.809 + 0.587i)14-s + (0.904 − 0.425i)15-s + (0.968 + 0.248i)16-s + (−0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.970 + 0.241i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ 0.970 + 0.241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.158597648 + 0.3875325377i\)
\(L(\frac12)\) \(\approx\) \(3.158597648 + 0.3875325377i\)
\(L(1)\) \(\approx\) \(1.978488843 + 0.06092396747i\)
\(L(1)\) \(\approx\) \(1.978488843 + 0.06092396747i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.998 + 0.0627i)T \)
3 \( 1 + (-0.368 - 0.929i)T \)
5 \( 1 + (0.0627 + 0.998i)T \)
7 \( 1 + (0.844 + 0.535i)T \)
11 \( 1 + (0.684 + 0.728i)T \)
13 \( 1 + (-0.535 - 0.844i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.968 - 0.248i)T \)
23 \( 1 + (0.637 - 0.770i)T \)
29 \( 1 + (0.844 - 0.535i)T \)
31 \( 1 + (0.535 - 0.844i)T \)
37 \( 1 + (-0.929 - 0.368i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
43 \( 1 + (-0.876 + 0.481i)T \)
47 \( 1 + (-0.876 - 0.481i)T \)
53 \( 1 + (0.125 + 0.992i)T \)
59 \( 1 + (-0.248 + 0.968i)T \)
61 \( 1 + (0.125 - 0.992i)T \)
67 \( 1 + (0.368 - 0.929i)T \)
71 \( 1 + (-0.929 + 0.368i)T \)
73 \( 1 + (-0.770 - 0.637i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (0.770 - 0.637i)T \)
89 \( 1 + (0.248 + 0.968i)T \)
97 \( 1 + (-0.992 - 0.125i)T \)
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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

−29.38649383684297752601180726270, −28.86406626403537472561159159637, −27.525702860708308020105884628755, −26.77078058902921371099534985345, −25.09600037187474181685790541132, −24.207191008379084900463198920447, −23.37336889891623325648452007163, −22.13854706706544690920011040301, −21.28192311746190006361761549034, −20.56023916991029679334506650895, −19.630420198180040330050135341831, −17.43135327195679494553876352647, −16.56023866441703365131334505116, −15.797076021723125329465565538817, −14.372112555312253783722374841964, −13.684283957808479383726708490301, −11.8860298793875887186260380284, −11.48639451980185246870025687271, −9.995655958871190445351104472553, −8.6591630213810792955702243938, −6.89593136129938864094398606181, −5.253918520975156874740639497356, −4.70834135780160551885288174707, −3.45983571501983651277737015196, −1.2633334583474031373924158176, 1.76524528537434114004961734014, 2.90019481439899593853720458684, 4.80912824766302432457595530706, 6.05385187632329316110398877826, 7.031047189412027588932965054612, 8.06114466469589645370426023193, 10.41380941368943194478886316635, 11.542932694252953137777143220417, 12.26348716945271561936957734898, 13.52391377915725682491992857825, 14.62672109206598652694124116592, 15.25886790234559461283036932911, 17.14716659797916846697466305135, 17.92426177651570934048674596005, 19.20578864129539061155424828379, 20.24969500371315255514304575899, 21.75106965779359464518987772791, 22.50783126907220563817419307200, 23.27956667569802627845311779479, 24.61947970639634266575037287144, 24.96346265959570954301898166061, 26.31568011759234397778210208863, 27.91367505990381372231410047050, 28.97521557082581231026815770145, 30.16939346868909182616207120927

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