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Properties

Label 1-101-101.22-r0-0-0
Degree $1$
Conductor $101$
Sign $-0.450 + 0.893i$
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  + (0.637 + 0.770i)2-s + (−0.535 − 0.844i)3-s + (−0.187 + 0.982i)4-s + (−0.637 + 0.770i)5-s + (0.309 − 0.951i)6-s + (−0.0627 + 0.998i)7-s + (−0.876 + 0.481i)8-s + (−0.425 + 0.904i)9-s − 10-s + (0.425 − 0.904i)11-s + (0.929 − 0.368i)12-s + (0.0627 + 0.998i)13-s + (−0.809 + 0.587i)14-s + (0.992 + 0.125i)15-s + (−0.929 − 0.368i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (0.637 + 0.770i)2-s + (−0.535 − 0.844i)3-s + (−0.187 + 0.982i)4-s + (−0.637 + 0.770i)5-s + (0.309 − 0.951i)6-s + (−0.0627 + 0.998i)7-s + (−0.876 + 0.481i)8-s + (−0.425 + 0.904i)9-s − 10-s + (0.425 − 0.904i)11-s + (0.929 − 0.368i)12-s + (0.0627 + 0.998i)13-s + (−0.809 + 0.587i)14-s + (0.992 + 0.125i)15-s + (−0.929 − 0.368i)16-s + (0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.450 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.450 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.450 + 0.893i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ -0.450 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4859656993 + 0.7890573946i\)
\(L(\frac12)\) \(\approx\) \(0.4859656993 + 0.7890573946i\)
\(L(1)\) \(\approx\) \(0.8439379608 + 0.5481786726i\)
\(L(1)\) \(\approx\) \(0.8439379608 + 0.5481786726i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.637 + 0.770i)T \)
3 \( 1 + (-0.535 - 0.844i)T \)
5 \( 1 + (-0.637 + 0.770i)T \)
7 \( 1 + (-0.0627 + 0.998i)T \)
11 \( 1 + (0.425 - 0.904i)T \)
13 \( 1 + (0.0627 + 0.998i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.929 + 0.368i)T \)
23 \( 1 + (0.968 + 0.248i)T \)
29 \( 1 + (-0.0627 - 0.998i)T \)
31 \( 1 + (0.0627 - 0.998i)T \)
37 \( 1 + (0.535 - 0.844i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (0.728 - 0.684i)T \)
47 \( 1 + (0.728 + 0.684i)T \)
53 \( 1 + (0.187 + 0.982i)T \)
59 \( 1 + (0.929 + 0.368i)T \)
61 \( 1 + (0.187 - 0.982i)T \)
67 \( 1 + (-0.535 + 0.844i)T \)
71 \( 1 + (0.535 + 0.844i)T \)
73 \( 1 + (-0.968 - 0.248i)T \)
79 \( 1 + (0.968 - 0.248i)T \)
83 \( 1 + (-0.968 + 0.248i)T \)
89 \( 1 + (0.929 - 0.368i)T \)
97 \( 1 + (-0.187 + 0.982i)T \)
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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

−29.502606901497445156599967201267, −28.54966270809360743469800831788, −27.465004509202513068776508446987, −27.22217406615699655511420596389, −25.34539562934147156586059348029, −23.82455416347764550755235043876, −23.092838418701600317604121487916, −22.4458292535321175451563297115, −20.957506626153343979613397881914, −20.37572450911845882047877188236, −19.5874278614521875062419384382, −17.800185955620528054702844595776, −16.69434929501268302781776790447, −15.536063601157193184540326424393, −14.59881612574662927397841838341, −13.04860372591849233744043221545, −12.14927201778239808067560896327, −11.013183466781095330957451589563, −10.116567859978001592565376960899, −8.93486950321751691081723018130, −6.937790131641340705945386389114, −5.161970456454513726245699249104, −4.44572723454846080743152135357, −3.33462325260433505507569997818, −0.86009151957033825889603939242, 2.49175638717995648124522578026, 4.05373489062476624057204375733, 5.8529609876562697234920275601, 6.45471291077171358945651719541, 7.729012494898740603376498308126, 8.76554785407493496957053369033, 11.167131218405078257802489735264, 11.89535461022352708047507147272, 12.96311459618298593676792339703, 14.231339610580352900070829661325, 15.14522010474179318329979266886, 16.39017837316304474719625655827, 17.338172720023823818744382625, 18.80261305043197592817633397732, 19.10238787609431923898285754596, 21.40330712337804676057474027910, 22.12316020370031596646757522408, 23.17172749856065062580346840570, 23.90597396807494701692123606649, 24.859198297493392571752668209988, 25.82361641218969474591174173440, 26.980956516648080627479599785614, 28.18168158613521821834932684707, 29.54340003335419864171631282432, 30.40859239306360295947787417093

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