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Properties

Label 1-101-101.64-r0-0-0
Degree $1$
Conductor $101$
Sign $0.118 + 0.992i$
Analytic cond. $0.469042$
Root an. cond. $0.469042$
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  + (0.929 + 0.368i)2-s + (0.637 + 0.770i)3-s + (0.728 + 0.684i)4-s + (−0.929 + 0.368i)5-s + (0.309 + 0.951i)6-s + (−0.968 − 0.248i)7-s + (0.425 + 0.904i)8-s + (−0.187 + 0.982i)9-s − 10-s + (0.187 − 0.982i)11-s + (−0.0627 + 0.998i)12-s + (0.968 − 0.248i)13-s + (−0.809 − 0.587i)14-s + (−0.876 − 0.481i)15-s + (0.0627 + 0.998i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.929 + 0.368i)2-s + (0.637 + 0.770i)3-s + (0.728 + 0.684i)4-s + (−0.929 + 0.368i)5-s + (0.309 + 0.951i)6-s + (−0.968 − 0.248i)7-s + (0.425 + 0.904i)8-s + (−0.187 + 0.982i)9-s − 10-s + (0.187 − 0.982i)11-s + (−0.0627 + 0.998i)12-s + (0.968 − 0.248i)13-s + (−0.809 − 0.587i)14-s + (−0.876 − 0.481i)15-s + (0.0627 + 0.998i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.118 + 0.992i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.118 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.275063620 + 1.131807295i\)
\(L(\frac12)\) \(\approx\) \(1.275063620 + 1.131807295i\)
\(L(1)\) \(\approx\) \(1.465329396 + 0.8087204767i\)
\(L(1)\) \(\approx\) \(1.465329396 + 0.8087204767i\)

Euler product

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

−30.04074442195792505845932629631, −28.71132175613816666065480158194, −28.082845074489802333861331989432, −26.259900800522960161628630316928, −25.25985989104873624007392528557, −24.38349829854483070077957900193, −23.16733134200049587471754148585, −22.85062025824894510969376474876, −21.037620305286556330166276028414, −20.21571789008647051687626642434, −19.31233206280193394637326466852, −18.6236137654826520282204502068, −16.57926500918170950159997392822, −15.393432454747008768245868017515, −14.59793911773085521709011222537, −13.11468721365955774154662370927, −12.58011751782504265692213669742, −11.62914774979391599740514940843, −9.94639722648312311849933994932, −8.47720406323968926104621082863, −7.09974553372570781477917607807, −6.06192386453912783548523235630, −4.14018879966315647308291653151, −3.21741033686692866780437613129, −1.57991063326660066993636806633, 3.19829171591913244362703715675, 3.45297539975516006389819734538, 4.978454853151309452952539943949, 6.5515704674278121718438126992, 7.78186583855571418790768328161, 9.00589652126441633235011217702, 10.7207909128075039532047384100, 11.61316398258865312853084361795, 13.276529259932201586511306655883, 13.98084116945850294164118269423, 15.401975424410481936716131000, 15.83547415857046345211309734381, 16.77260409618962508643765390989, 18.927308373552359382160621507587, 19.86343986721082375701146732665, 20.80764056354762298450412607172, 22.0186690370677510989420458262, 22.73980976937774319162317525052, 23.71876430367955750153153214129, 25.04782999862423932405755684624, 26.02592198465583199408754302430, 26.704989818436354490716569559613, 27.891403850648489261601487128297, 29.45312504390297316597931048884, 30.441886915661107461138645558951

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