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Properties

Label 1-101-101.23-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: $\chi_{101} (23, \cdot )$
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

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

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