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Properties

Label 1-101-101.35-r1-0-0
Degree $1$
Conductor $101$
Sign $0.498 - 0.866i$
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  + (−0.481 + 0.876i)2-s + (0.125 − 0.992i)3-s + (−0.535 − 0.844i)4-s + (0.876 − 0.481i)5-s + (0.809 + 0.587i)6-s + (0.982 − 0.187i)7-s + (0.998 − 0.0627i)8-s + (−0.968 − 0.248i)9-s + i·10-s + (−0.248 + 0.968i)11-s + (−0.904 + 0.425i)12-s + (0.187 − 0.982i)13-s + (−0.309 + 0.951i)14-s + (−0.368 − 0.929i)15-s + (−0.425 + 0.904i)16-s + (0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−0.481 + 0.876i)2-s + (0.125 − 0.992i)3-s + (−0.535 − 0.844i)4-s + (0.876 − 0.481i)5-s + (0.809 + 0.587i)6-s + (0.982 − 0.187i)7-s + (0.998 − 0.0627i)8-s + (−0.968 − 0.248i)9-s + i·10-s + (−0.248 + 0.968i)11-s + (−0.904 + 0.425i)12-s + (0.187 − 0.982i)13-s + (−0.309 + 0.951i)14-s + (−0.368 − 0.929i)15-s + (−0.425 + 0.904i)16-s + (0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.498 - 0.866i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ 0.498 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.348820615 - 0.7798931078i\)
\(L(\frac12)\) \(\approx\) \(1.348820615 - 0.7798931078i\)
\(L(1)\) \(\approx\) \(1.047168244 - 0.1863318602i\)
\(L(1)\) \(\approx\) \(1.047168244 - 0.1863318602i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.481 + 0.876i)T \)
3 \( 1 + (0.125 - 0.992i)T \)
5 \( 1 + (0.876 - 0.481i)T \)
7 \( 1 + (0.982 - 0.187i)T \)
11 \( 1 + (-0.248 + 0.968i)T \)
13 \( 1 + (0.187 - 0.982i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.425 - 0.904i)T \)
23 \( 1 + (-0.728 + 0.684i)T \)
29 \( 1 + (0.982 + 0.187i)T \)
31 \( 1 + (-0.187 - 0.982i)T \)
37 \( 1 + (-0.992 + 0.125i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (0.637 - 0.770i)T \)
47 \( 1 + (0.637 + 0.770i)T \)
53 \( 1 + (-0.844 - 0.535i)T \)
59 \( 1 + (-0.904 - 0.425i)T \)
61 \( 1 + (-0.844 + 0.535i)T \)
67 \( 1 + (-0.125 - 0.992i)T \)
71 \( 1 + (-0.992 - 0.125i)T \)
73 \( 1 + (0.684 + 0.728i)T \)
79 \( 1 + (0.728 + 0.684i)T \)
83 \( 1 + (-0.684 + 0.728i)T \)
89 \( 1 + (0.904 - 0.425i)T \)
97 \( 1 + (0.535 + 0.844i)T \)
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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

−29.68256133331472432477679918712, −28.68090003300983594672359438502, −27.78616763614396872204497309566, −26.761385717781373181756513892185, −26.16320150852033457627480136626, −25.00518165656200550209111292863, −23.268527306577139039140097767567, −21.85080426039297430608398081288, −21.362001065670229766269013557261, −20.783781184862270805625660070206, −19.25262363072307576552683673237, −18.27738362131993518129438379742, −17.166625667650779443685807391182, −16.270988603229104148496597090181, −14.42160712986767596760261115356, −13.92765528763763859125380523165, −12.08758951269885878469562730038, −10.86936042796236752517593293431, −10.29576300745636667855914592086, −9.01157253842014954298917099983, −8.12543914072405711690265107805, −5.901372143689161581353395712946, −4.45989309696533420646543868532, −3.097210675696574000337406908533, −1.72658113911029856922736279333, 0.8480102679133067432636434489, 2.08325645070181145737565308684, 4.91658849429405052185618481817, 5.867212515927193345005626772269, 7.2845621832974098401862378546, 8.112408524729074008287698738339, 9.2995118001812708664896591334, 10.61372219691633574781200495637, 12.362019077437747188569739801473, 13.55243768579460440843358958079, 14.30763932842879186360761475474, 15.54570387512193782413597372362, 17.262808675589973615199241040214, 17.60474792722161992677963577897, 18.44828255561646598535253517193, 19.90148740875159601254335841623, 20.80530301760819424748084098927, 22.57054398376457352619374496966, 23.69486424056809087713671816092, 24.37941560223796588497327378282, 25.42893119389360378531019606746, 25.81111974542471587744409154724, 27.54758051240237094012168613944, 28.23759365198639414415349845091, 29.44544196404668729623351057270

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