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Properties

Label 1-101-101.53-r1-0-0
Degree $1$
Conductor $101$
Sign $0.856 + 0.515i$
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  + (0.125 + 0.992i)2-s + (0.684 − 0.728i)3-s + (−0.968 + 0.248i)4-s + (−0.992 − 0.125i)5-s + (0.809 + 0.587i)6-s + (0.904 + 0.425i)7-s + (−0.368 − 0.929i)8-s + (−0.0627 − 0.998i)9-s − i·10-s + (0.998 − 0.0627i)11-s + (−0.481 + 0.876i)12-s + (0.425 + 0.904i)13-s + (−0.309 + 0.951i)14-s + (−0.770 + 0.637i)15-s + (0.876 − 0.481i)16-s + (0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.125 + 0.992i)2-s + (0.684 − 0.728i)3-s + (−0.968 + 0.248i)4-s + (−0.992 − 0.125i)5-s + (0.809 + 0.587i)6-s + (0.904 + 0.425i)7-s + (−0.368 − 0.929i)8-s + (−0.0627 − 0.998i)9-s − i·10-s + (0.998 − 0.0627i)11-s + (−0.481 + 0.876i)12-s + (0.425 + 0.904i)13-s + (−0.309 + 0.951i)14-s + (−0.770 + 0.637i)15-s + (0.876 − 0.481i)16-s + (0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.856 + 0.515i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ 0.856 + 0.515i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.990496601 + 0.5531527292i\)
\(L(\frac12)\) \(\approx\) \(1.990496601 + 0.5531527292i\)
\(L(1)\) \(\approx\) \(1.310919743 + 0.3340005020i\)
\(L(1)\) \(\approx\) \(1.310919743 + 0.3340005020i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.125 + 0.992i)T \)
3 \( 1 + (0.684 - 0.728i)T \)
5 \( 1 + (-0.992 - 0.125i)T \)
7 \( 1 + (0.904 + 0.425i)T \)
11 \( 1 + (0.998 - 0.0627i)T \)
13 \( 1 + (0.425 + 0.904i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.876 + 0.481i)T \)
23 \( 1 + (0.187 - 0.982i)T \)
29 \( 1 + (0.904 - 0.425i)T \)
31 \( 1 + (-0.425 + 0.904i)T \)
37 \( 1 + (0.728 - 0.684i)T \)
41 \( 1 + (-0.587 + 0.809i)T \)
43 \( 1 + (-0.535 - 0.844i)T \)
47 \( 1 + (-0.535 + 0.844i)T \)
53 \( 1 + (-0.248 + 0.968i)T \)
59 \( 1 + (-0.481 - 0.876i)T \)
61 \( 1 + (-0.248 - 0.968i)T \)
67 \( 1 + (-0.684 - 0.728i)T \)
71 \( 1 + (0.728 + 0.684i)T \)
73 \( 1 + (0.982 + 0.187i)T \)
79 \( 1 + (-0.187 - 0.982i)T \)
83 \( 1 + (-0.982 + 0.187i)T \)
89 \( 1 + (0.481 - 0.876i)T \)
97 \( 1 + (0.968 - 0.248i)T \)
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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

−29.99071270841209841714869499574, −28.15198892158290607939101482993, −27.417650466828579042178968456223, −27.03793989404435216438980650731, −25.69830777186295867706665740408, −24.16033027668624830297919699611, −23.04013309930807879062078282730, −22.06040730693709289179922684974, −20.987006707829324722710912222729, −20.04084530380393345193175579739, −19.55694629249541493545999829326, −18.1844237567815831937082533311, −16.829147264524068175286339323049, −15.2497830519619839449812697802, −14.521052983857279586200844746048, −13.41627654425433895259577841100, −11.837487681426458288638187332126, −11.029526138093325391007816269809, −9.93801795192573621555336213380, −8.60451316127172127762871447107, −7.727637691868041746069898478223, −5.14705613810112457686877640995, −3.990786743651080037962260235972, −3.19463335338833943405496593559, −1.23995839979314684101512346840, 1.13935846359545651884988348010, 3.42296276767922940093052203568, 4.68333254873790358399603895268, 6.40149528554953548986511253688, 7.512250388545416831665240877084, 8.38206792953602654813036295550, 9.24828271771426117892596702120, 11.67609317667166170741485231830, 12.41383011623794546792595881647, 14.12018327595377806787472833176, 14.46750671594210800104726031075, 15.727350612255040427373552878678, 16.84225705689129643552389958079, 18.270901033427980261225908174462, 18.85522720520187598970503560738, 20.14763213880289300108750808176, 21.41032102916061190696691090118, 22.9265513638171847905614924852, 23.73744819733892671211249245395, 24.65193811976607860226905129932, 25.21476205428597924808925729053, 26.68887627499297853240417489958, 27.221541779372304469861314914005, 28.48974020829642489706154044330, 30.37700004607146010048568020096

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