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Properties

Label 1-101-101.79-r0-0-0
Degree $1$
Conductor $101$
Sign $0.379 + 0.925i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.379 + 0.925i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.379 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5779762767 + 0.3875442234i\)
\(L(\frac12)\) \(\approx\) \(0.5779762767 + 0.3875442234i\)
\(L(1)\) \(\approx\) \(0.7488554777 + 0.1663532254i\)
\(L(1)\) \(\approx\) \(0.7488554777 + 0.1663532254i\)

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

−29.49818755399961143471278403079, −28.52345950344027674905335409145, −27.53842736999324535120650458767, −26.58081742467515501467125986496, −25.07128028839637796085117580393, −24.93464361822240077597599008904, −23.802086973006814374757036836740, −22.96392200702839555772026943056, −21.0410202025471343252461759857, −19.85454352925943174575533358610, −18.97874400756205548327227502782, −18.23286082209189930155524179409, −16.994680710157111593744114928708, −15.72242082515050675516263812686, −15.0014064500115539662823513416, −13.57685078887230509102582983451, −12.5238534163556636697498533033, −11.179085472736783117487197627111, −9.27474863895006716613850070856, −8.42981447248154039506387639387, −7.75210966619697473645413312359, −6.233940818916591610878896402196, −5.06072404129712237255540300831, −2.782616690923834851740450428585, −0.84852777437222425490881910015, 2.17612730220879115345294408646, 3.67545715208702585212029953350, 4.36010644423600867891044316970, 7.09856628379266177936782919506, 8.039880168899211922981871753697, 9.38891473989988334031764504936, 10.54529024730768001143213537718, 11.00989254976709079970387303068, 12.61469081179841383839502171770, 14.04727345521741030455988236424, 15.11246398208018336856250398928, 16.41727553000392806941186476931, 17.35401197525382210555469530107, 18.89071443032221708029837914445, 19.57246080253545355561615739880, 20.601835436346860604305674810756, 21.411803913050841210758235692476, 22.59763568460843782151821616379, 23.55986289566233967907885322523, 25.77556803957273455855149045198, 26.05913997426234881262561917810, 27.106781095334281835035661888957, 27.742511643986319670747269077711, 28.96687082235841715263682722403, 30.2573829894386296380687962396

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