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Properties

Label 1-101-101.3-r1-0-0
Degree $1$
Conductor $101$
Sign $0.421 - 0.906i$
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  + (−0.368 − 0.929i)2-s + (−0.770 − 0.637i)3-s + (−0.728 + 0.684i)4-s + (−0.929 − 0.368i)5-s + (−0.309 + 0.951i)6-s + (0.248 + 0.968i)7-s + (0.904 + 0.425i)8-s + (0.187 + 0.982i)9-s + i·10-s + (0.982 − 0.187i)11-s + (0.998 − 0.0627i)12-s + (−0.968 − 0.248i)13-s + (0.809 − 0.587i)14-s + (0.481 + 0.876i)15-s + (0.0627 − 0.998i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.368 − 0.929i)2-s + (−0.770 − 0.637i)3-s + (−0.728 + 0.684i)4-s + (−0.929 − 0.368i)5-s + (−0.309 + 0.951i)6-s + (0.248 + 0.968i)7-s + (0.904 + 0.425i)8-s + (0.187 + 0.982i)9-s + i·10-s + (0.982 − 0.187i)11-s + (0.998 − 0.0627i)12-s + (−0.968 − 0.248i)13-s + (0.809 − 0.587i)14-s + (0.481 + 0.876i)15-s + (0.0627 − 0.998i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.421 - 0.906i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ 0.421 - 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6265063679 - 0.3996794898i\)
\(L(\frac12)\) \(\approx\) \(0.6265063679 - 0.3996794898i\)
\(L(1)\) \(\approx\) \(0.5361099544 - 0.2895516876i\)
\(L(1)\) \(\approx\) \(0.5361099544 - 0.2895516876i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.368 - 0.929i)T \)
3 \( 1 + (-0.770 - 0.637i)T \)
5 \( 1 + (-0.929 - 0.368i)T \)
7 \( 1 + (0.248 + 0.968i)T \)
11 \( 1 + (0.982 - 0.187i)T \)
13 \( 1 + (-0.968 - 0.248i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.0627 + 0.998i)T \)
23 \( 1 + (-0.535 + 0.844i)T \)
29 \( 1 + (0.248 - 0.968i)T \)
31 \( 1 + (0.968 - 0.248i)T \)
37 \( 1 + (-0.637 - 0.770i)T \)
41 \( 1 + (0.951 + 0.309i)T \)
43 \( 1 + (0.992 - 0.125i)T \)
47 \( 1 + (0.992 + 0.125i)T \)
53 \( 1 + (0.684 - 0.728i)T \)
59 \( 1 + (0.998 + 0.0627i)T \)
61 \( 1 + (0.684 + 0.728i)T \)
67 \( 1 + (0.770 - 0.637i)T \)
71 \( 1 + (-0.637 + 0.770i)T \)
73 \( 1 + (0.844 + 0.535i)T \)
79 \( 1 + (0.535 + 0.844i)T \)
83 \( 1 + (-0.844 + 0.535i)T \)
89 \( 1 + (-0.998 + 0.0627i)T \)
97 \( 1 + (0.728 - 0.684i)T \)
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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

−29.76331897817255820406244210464, −28.32565967679338416904448910252, −27.54062683686528370516444240295, −26.7300305538297744148828675086, −26.15865388380552960988694764103, −24.29214466767655972693952510342, −23.738918445518445645216753488145, −22.66575672684645984528255326527, −21.9960419341338061458751697220, −20.10116766413048208520547961520, −19.24956463929752183264168747593, −17.682670815818231136547072223616, −17.05881723854953285448088162685, −16.06639862431293009175835276137, −15.02405532930693315096049103249, −14.21640281363606702817897943111, −12.32138335425406548342328029351, −10.99447177836538575384801466503, −10.117015218139323086014858328043, −8.73533227856063061500814307210, −7.23676880928293358885851047845, −6.505330601412869122166466596925, −4.70641289472066855591690129105, −4.02288427313489577555909102003, −0.68853693257504646576266964685, 0.78427473699542099157769068894, 2.35219996784977520852111901796, 4.18372873101104936264627520718, 5.48727917544509992603883929849, 7.36353071228879976269788427625, 8.38919773140465037651665740307, 9.70618619136353638811190420282, 11.37311968918482111075016816456, 11.91176744035460071208163593835, 12.58063434043903324030121008280, 14.1156344663327889150667375676, 15.83859010839865630181612124787, 17.020223361118152265579604084913, 17.94931444075290777367207816723, 19.097334715723292245590327446543, 19.61086478349535011860392283213, 21.03826920147123890798329126942, 22.315774338389892585045667840460, 22.836177321512664055067981431734, 24.33684400151993562963797268052, 25.06331719733095816097564802614, 26.99804334564512070019269808005, 27.60367285116527978167136815081, 28.392497456373057183421891704318, 29.41564242006042524380267382898

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