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Properties

Label 1-101-101.52-r0-0-0
Degree $1$
Conductor $101$
Sign $0.650 - 0.759i$
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  + (−0.425 − 0.904i)2-s + (0.876 − 0.481i)3-s + (−0.637 + 0.770i)4-s + (−0.425 + 0.904i)5-s + (−0.809 − 0.587i)6-s + (0.728 + 0.684i)7-s + (0.968 + 0.248i)8-s + (0.535 − 0.844i)9-s + 10-s + (0.535 − 0.844i)11-s + (−0.187 + 0.982i)12-s + (0.728 − 0.684i)13-s + (0.309 − 0.951i)14-s + (0.0627 + 0.998i)15-s + (−0.187 − 0.982i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.425 − 0.904i)2-s + (0.876 − 0.481i)3-s + (−0.637 + 0.770i)4-s + (−0.425 + 0.904i)5-s + (−0.809 − 0.587i)6-s + (0.728 + 0.684i)7-s + (0.968 + 0.248i)8-s + (0.535 − 0.844i)9-s + 10-s + (0.535 − 0.844i)11-s + (−0.187 + 0.982i)12-s + (0.728 − 0.684i)13-s + (0.309 − 0.951i)14-s + (0.0627 + 0.998i)15-s + (−0.187 − 0.982i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.650 - 0.759i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.650 - 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9551649212 - 0.4396264903i\)
\(L(\frac12)\) \(\approx\) \(0.9551649212 - 0.4396264903i\)
\(L(1)\) \(\approx\) \(0.9959773437 - 0.3722989974i\)
\(L(1)\) \(\approx\) \(0.9959773437 - 0.3722989974i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.425 - 0.904i)T \)
3 \( 1 + (0.876 - 0.481i)T \)
5 \( 1 + (-0.425 + 0.904i)T \)
7 \( 1 + (0.728 + 0.684i)T \)
11 \( 1 + (0.535 - 0.844i)T \)
13 \( 1 + (0.728 - 0.684i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.187 + 0.982i)T \)
23 \( 1 + (-0.992 + 0.125i)T \)
29 \( 1 + (0.728 - 0.684i)T \)
31 \( 1 + (0.728 + 0.684i)T \)
37 \( 1 + (0.876 + 0.481i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.929 - 0.368i)T \)
47 \( 1 + (-0.929 + 0.368i)T \)
53 \( 1 + (-0.637 - 0.770i)T \)
59 \( 1 + (-0.187 - 0.982i)T \)
61 \( 1 + (-0.637 + 0.770i)T \)
67 \( 1 + (0.876 + 0.481i)T \)
71 \( 1 + (0.876 - 0.481i)T \)
73 \( 1 + (-0.992 + 0.125i)T \)
79 \( 1 + (-0.992 - 0.125i)T \)
83 \( 1 + (-0.992 - 0.125i)T \)
89 \( 1 + (-0.187 + 0.982i)T \)
97 \( 1 + (-0.637 + 0.770i)T \)
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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

−30.437413268020117471706322444081, −28.426026827921583339276841886251, −27.74715965563351308493690122015, −26.83915969452806588010868515851, −25.98747454339192063217466802445, −24.88377648457206429978587428535, −24.117758068580405505144385155157, −23.134403876452488077283685689, −21.5863002204447955525452098807, −20.14741324287322152693149557380, −19.872119309848946424534023279657, −18.26137630202024561569179362795, −17.0817145278762118827706513623, −16.10377929292805425131602618274, −15.23265801367492699300380250278, −14.11108002364492683850428327618, −13.2595214350267231189368574061, −11.31384180456045251046111929559, −9.81411252034858673190189055289, −8.84769707764340526070832111275, −8.0136769268281372915777791525, −6.84489338033224297539241451118, −4.69865037997089211833961795222, −4.27187918063757632404081594817, −1.57719277422445822209631331114, 1.67398462177915119821629795452, 2.96044529334050869108680335941, 3.97879125254521148782511952395, 6.35018477905551529959052890008, 8.15581414246114727331408521307, 8.400084864653265839138473945498, 10.05655046474211492749576709144, 11.27629173914567358197252331560, 12.184686682645804013326380609984, 13.5720010275356055028477872166, 14.48650507056526841706208356304, 15.68816005757535940912843722344, 17.6343316115120205825796879238, 18.44257011198151666576587178379, 19.17204791292739327718313410549, 20.11204020560807005115911284747, 21.25279107500101208592164336280, 22.127041907157318573776113943125, 23.46584603063015266575305056218, 24.82773248157988507915864072880, 25.8094008123298297379647885221, 26.884905032531207898727716911672, 27.46974014342236221757243663821, 28.86052878166721126862312736390, 30.20511843173638336231640099022

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