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Properties

Label 1-101-101.37-r0-0-0
Degree $1$
Conductor $101$
Sign $0.909 - 0.415i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.909 - 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5189954876 - 0.1130466550i\)
\(L(\frac12)\) \(\approx\) \(0.5189954876 - 0.1130466550i\)
\(L(1)\) \(\approx\) \(0.5758511374 - 0.1221900108i\)
\(L(1)\) \(\approx\) \(0.5758511374 - 0.1221900108i\)

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

−29.651777141941381965321469531842, −28.42924456878776832403964488275, −27.82701632053794914135487566758, −26.93838650514046233140927316491, −26.378171676356549147337865017344, −24.7046087695367390034754509425, −23.740045246999010008661665108271, −23.10715745470467020066265482071, −21.201222010027121535685110439811, −20.66073386686149517770922530476, −19.267558556794756118835141055786, −18.22650062743173761926711204243, −16.98786010216074065649158104960, −16.309966122178437058636475037534, −15.3941229751505741308538396679, −14.314925190387145743981873092727, −12.087587859957909646848813317096, −11.096746148520746687513550218394, −10.453178393244753693994301914784, −8.70058953938397431840073931035, −8.11752900541182488472767468189, −6.40317806933558658757584070306, −5.14838854545390266973742410581, −3.72075195618071259158893649800, −1.048523598876213816904366841078, 1.20655933745250725896553770069, 2.79742560101044105431506417591, 4.81689185843047189145850214504, 6.73522159047708425565338010208, 7.61749763658378160482025155403, 8.5320475891576527340736457338, 10.35083923449345237519906942002, 11.52756510219758332716958959908, 11.84285221924402805223806967423, 13.32640560089047143286390883482, 15.16625538716511757643844201107, 16.13345067716378369113813101689, 17.634502598919776332331349773470, 18.07201387739227062647676741970, 19.096196650530279753388592310177, 20.10153064288481737836674306021, 21.2474770283058861863032604084, 22.722922972922348450504417815932, 23.61039430033221378991757855614, 24.76222284953970574395650408323, 25.71351288186563703198713406079, 27.097895141583848904383579640197, 27.88578530830460870643222399215, 28.51329048427038692892622661408, 29.89743747986349652059411616694

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