Properties

Label 1-101-101.82-r0-0-0
Degree $1$
Conductor $101$
Sign $0.965 - 0.261i$
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  + (−0.968 + 0.248i)2-s + (−0.0627 − 0.998i)3-s + (0.876 − 0.481i)4-s + (0.968 + 0.248i)5-s + (0.309 + 0.951i)6-s + (0.637 + 0.770i)7-s + (−0.728 + 0.684i)8-s + (−0.992 + 0.125i)9-s − 10-s + (0.992 − 0.125i)11-s + (−0.535 − 0.844i)12-s + (−0.637 + 0.770i)13-s + (−0.809 − 0.587i)14-s + (0.187 − 0.982i)15-s + (0.535 − 0.844i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.968 + 0.248i)2-s + (−0.0627 − 0.998i)3-s + (0.876 − 0.481i)4-s + (0.968 + 0.248i)5-s + (0.309 + 0.951i)6-s + (0.637 + 0.770i)7-s + (−0.728 + 0.684i)8-s + (−0.992 + 0.125i)9-s − 10-s + (0.992 − 0.125i)11-s + (−0.535 − 0.844i)12-s + (−0.637 + 0.770i)13-s + (−0.809 − 0.587i)14-s + (0.187 − 0.982i)15-s + (0.535 − 0.844i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.965 - 0.261i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.965 - 0.261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8019901440 - 0.1066847071i\)
\(L(\frac12)\) \(\approx\) \(0.8019901440 - 0.1066847071i\)
\(L(1)\) \(\approx\) \(0.8294047672 - 0.07928977627i\)
\(L(1)\) \(\approx\) \(0.8294047672 - 0.07928977627i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.968 + 0.248i)T \)
3 \( 1 + (-0.0627 - 0.998i)T \)
5 \( 1 + (0.968 + 0.248i)T \)
7 \( 1 + (0.637 + 0.770i)T \)
11 \( 1 + (0.992 - 0.125i)T \)
13 \( 1 + (-0.637 + 0.770i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.535 + 0.844i)T \)
23 \( 1 + (-0.929 - 0.368i)T \)
29 \( 1 + (0.637 - 0.770i)T \)
31 \( 1 + (-0.637 - 0.770i)T \)
37 \( 1 + (0.0627 - 0.998i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (-0.425 + 0.904i)T \)
47 \( 1 + (-0.425 - 0.904i)T \)
53 \( 1 + (-0.876 - 0.481i)T \)
59 \( 1 + (-0.535 + 0.844i)T \)
61 \( 1 + (-0.876 + 0.481i)T \)
67 \( 1 + (-0.0627 + 0.998i)T \)
71 \( 1 + (0.0627 + 0.998i)T \)
73 \( 1 + (0.929 + 0.368i)T \)
79 \( 1 + (-0.929 + 0.368i)T \)
83 \( 1 + (0.929 - 0.368i)T \)
89 \( 1 + (-0.535 - 0.844i)T \)
97 \( 1 + (0.876 - 0.481i)T \)
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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

−29.75813887381949264651955655487, −28.68603775394162642284965331694, −27.72941064467411038136815299654, −27.05243054965938159192745681532, −25.99221024124193769027699505892, −25.1755927608819828651935452163, −23.95262153697519072440364945904, −22.08922967019732247047170740675, −21.52894977186978550709905572442, −20.253549575880198360887050435964, −19.87250725916263630929873844883, −17.84965922481074626517158786460, −17.28439350336669109817831803058, −16.51982211888708126869904069257, −15.10194114242864295194835182090, −14.00981315423973052967700498786, −12.27358624473157680390867117717, −10.940894436905968007285601764977, −10.10594473608073405893570945838, −9.261002979260149847254654260943, −8.05076633149850038104616347639, −6.4273937845822224684489870411, −4.905543718827292590332937416562, −3.2920134646250564681212748596, −1.49722932797107255331806266828, 1.55772993409960496309542313634, 2.45751576251356078436459829679, 5.52097611777216769352538042691, 6.43139294996320425084473147494, 7.567151470473614474109977668054, 8.83962119242232360229057005481, 9.78454690465211897409009583632, 11.457505225392186674843144283447, 12.13778598458060350659821624434, 14.07041363284952937024498467633, 14.57908568291013107227649084052, 16.44911102103000448288006117250, 17.39024149293064231650982389257, 18.25049042200612126908963356105, 18.88765004506839749096765217080, 20.12284272510009854142213422268, 21.35109043141100144937565077918, 22.63695814651163661565153714560, 24.29139946620050681578823905545, 24.75230985331579680441510905165, 25.49665625747873971001878014587, 26.69665577987264292869643875089, 27.897822814360236019489758198753, 28.899752310808414257316546218309, 29.6079224962002301891047964676

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