Properties

Label 1-101-101.20-r0-0-0
Degree $1$
Conductor $101$
Sign $-0.228 + 0.973i$
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  + (−0.0627 + 0.998i)2-s + (0.929 − 0.368i)3-s + (−0.992 − 0.125i)4-s + (0.0627 + 0.998i)5-s + (0.309 + 0.951i)6-s + (−0.535 + 0.844i)7-s + (0.187 − 0.982i)8-s + (0.728 − 0.684i)9-s − 10-s + (−0.728 + 0.684i)11-s + (−0.968 + 0.248i)12-s + (0.535 + 0.844i)13-s + (−0.809 − 0.587i)14-s + (0.425 + 0.904i)15-s + (0.968 + 0.248i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.0627 + 0.998i)2-s + (0.929 − 0.368i)3-s + (−0.992 − 0.125i)4-s + (0.0627 + 0.998i)5-s + (0.309 + 0.951i)6-s + (−0.535 + 0.844i)7-s + (0.187 − 0.982i)8-s + (0.728 − 0.684i)9-s − 10-s + (−0.728 + 0.684i)11-s + (−0.968 + 0.248i)12-s + (0.535 + 0.844i)13-s + (−0.809 − 0.587i)14-s + (0.425 + 0.904i)15-s + (0.968 + 0.248i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.228 + 0.973i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ -0.228 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7018514899 + 0.8860146131i\)
\(L(\frac12)\) \(\approx\) \(0.7018514899 + 0.8860146131i\)
\(L(1)\) \(\approx\) \(0.9437244725 + 0.6569634664i\)
\(L(1)\) \(\approx\) \(0.9437244725 + 0.6569634664i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.0627 + 0.998i)T \)
3 \( 1 + (0.929 - 0.368i)T \)
5 \( 1 + (0.0627 + 0.998i)T \)
7 \( 1 + (-0.535 + 0.844i)T \)
11 \( 1 + (-0.728 + 0.684i)T \)
13 \( 1 + (0.535 + 0.844i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.968 - 0.248i)T \)
23 \( 1 + (-0.637 + 0.770i)T \)
29 \( 1 + (-0.535 - 0.844i)T \)
31 \( 1 + (0.535 - 0.844i)T \)
37 \( 1 + (-0.929 - 0.368i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (0.876 - 0.481i)T \)
47 \( 1 + (0.876 + 0.481i)T \)
53 \( 1 + (0.992 - 0.125i)T \)
59 \( 1 + (-0.968 - 0.248i)T \)
61 \( 1 + (0.992 + 0.125i)T \)
67 \( 1 + (0.929 + 0.368i)T \)
71 \( 1 + (-0.929 + 0.368i)T \)
73 \( 1 + (0.637 - 0.770i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (0.637 + 0.770i)T \)
89 \( 1 + (-0.968 + 0.248i)T \)
97 \( 1 + (-0.992 - 0.125i)T \)
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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

−29.63775497387705771306366025718, −28.586573459772854627206629957449, −27.626086021799915351382836829534, −26.5940392061574246520655202442, −25.835437967775035439104178930866, −24.39245374442239506624549394092, −23.25731509449123422838269507372, −21.92550300194137821449520626488, −20.85794577200559726741855831306, −20.273180604650449650775507296757, −19.487744406137639812892955075086, −18.281896587237269406987664414429, −16.79367422858252481876892446928, −15.81971625285120288478410522370, −14.097420883433437851426391377890, −13.2933851101760202435921153966, −12.5206948072238173767381028823, −10.66078580096094718819762387348, −9.95708554753131495998934957772, −8.66784846265423089053584259592, −7.92305426663896516829741416422, −5.39228498218468163159097394216, −4.01261879324965408558169516657, −3.07197444089999428263097476365, −1.24400886473011628264011861198, 2.38520664493410107666822518033, 3.765767393157776770380705459441, 5.675792259072029450617177980701, 6.943065729626711481856710144608, 7.70675917117427318241830727020, 9.16695507811780738215501717291, 9.90091339782125292613054449298, 11.99177776861746700686502124892, 13.45684030824496566436591560798, 14.11929029678841474966335170652, 15.38631925407303018177255154796, 15.813367120674464349974854898567, 17.77339055372194290248181068978, 18.59158479531885724681769533638, 19.14009086180173762639510189610, 20.836018321764487521577316507339, 22.11003532960795565128809730535, 23.050497819478099374282530363706, 24.20173379518938344722219174924, 25.2740906950767907306679692881, 25.99122204920349223468203607762, 26.49868996909223345448481672508, 27.90420027417596832516378354482, 29.206797755849976935316046180795, 30.66950217515616332817813779958

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