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Properties

Label 1-101-101.47-r0-0-0
Degree $1$
Conductor $101$
Sign $0.985 - 0.170i$
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  + (−0.876 − 0.481i)2-s + (0.992 + 0.125i)3-s + (0.535 + 0.844i)4-s + (0.876 − 0.481i)5-s + (−0.809 − 0.587i)6-s + (0.187 + 0.982i)7-s + (−0.0627 − 0.998i)8-s + (0.968 + 0.248i)9-s − 10-s + (−0.968 − 0.248i)11-s + (0.425 + 0.904i)12-s + (−0.187 + 0.982i)13-s + (0.309 − 0.951i)14-s + (0.929 − 0.368i)15-s + (−0.425 + 0.904i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.876 − 0.481i)2-s + (0.992 + 0.125i)3-s + (0.535 + 0.844i)4-s + (0.876 − 0.481i)5-s + (−0.809 − 0.587i)6-s + (0.187 + 0.982i)7-s + (−0.0627 − 0.998i)8-s + (0.968 + 0.248i)9-s − 10-s + (−0.968 − 0.248i)11-s + (0.425 + 0.904i)12-s + (−0.187 + 0.982i)13-s + (0.309 − 0.951i)14-s + (0.929 − 0.368i)15-s + (−0.425 + 0.904i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.985 - 0.170i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.985 - 0.170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.036170861 - 0.08912721614i\)
\(L(\frac12)\) \(\approx\) \(1.036170861 - 0.08912721614i\)
\(L(1)\) \(\approx\) \(1.039136687 - 0.1002765246i\)
\(L(1)\) \(\approx\) \(1.039136687 - 0.1002765246i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.876 - 0.481i)T \)
3 \( 1 + (0.992 + 0.125i)T \)
5 \( 1 + (0.876 - 0.481i)T \)
7 \( 1 + (0.187 + 0.982i)T \)
11 \( 1 + (-0.968 - 0.248i)T \)
13 \( 1 + (-0.187 + 0.982i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.425 - 0.904i)T \)
23 \( 1 + (0.728 - 0.684i)T \)
29 \( 1 + (0.187 - 0.982i)T \)
31 \( 1 + (-0.187 - 0.982i)T \)
37 \( 1 + (-0.992 + 0.125i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (-0.637 + 0.770i)T \)
47 \( 1 + (-0.637 - 0.770i)T \)
53 \( 1 + (-0.535 + 0.844i)T \)
59 \( 1 + (0.425 - 0.904i)T \)
61 \( 1 + (-0.535 - 0.844i)T \)
67 \( 1 + (0.992 - 0.125i)T \)
71 \( 1 + (-0.992 - 0.125i)T \)
73 \( 1 + (-0.728 + 0.684i)T \)
79 \( 1 + (0.728 + 0.684i)T \)
83 \( 1 + (-0.728 - 0.684i)T \)
89 \( 1 + (0.425 + 0.904i)T \)
97 \( 1 + (0.535 + 0.844i)T \)
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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

−29.60320335306546269157295868169, −29.18346467327602902555649619943, −27.37893070000854515444800601349, −26.71720651074206447674718824708, −25.75594096379849743331670154157, −25.1772980263647079508414897825, −24.0657016341470930961331346283, −22.88966929162386177310124817677, −21.074385336966127408409310456563, −20.38082352739017891622434861423, −19.317009264774369876017386304858, −18.122617015338931044378766695436, −17.53140873854603361900754508739, −16.03305465666497435642287827835, −14.91781245726843880810550215830, −14.02372288487951248650602057034, −13.01069336128386061293640310901, −10.650834646149841242095881296970, −10.14515943250716218450500640376, −8.876691302067114625225399325915, −7.631136877112663658714059111477, −6.87134288673926634587072616973, −5.18103551623355675409022429696, −2.99102510638002305528981887147, −1.6420593964706382511743934773, 1.95272867382749856251534942395, 2.65799355751165519662998722068, 4.620777412372634187980116010017, 6.53688089985247246787870340358, 8.1993369770296009419532719439, 8.935104078737103277855569748407, 9.75596024520057926150110477613, 11.08186724138045390254850907989, 12.66470354364917880296035918427, 13.43606911566493642433952336245, 15.03765326672115705727269818923, 16.06713699149771563451229744550, 17.35073262379491553290346444870, 18.480851289959087489960057436097, 19.23128407783151928179409035252, 20.47894509260579252478442801179, 21.364452989567056323458283644578, 21.76692348931841629733462163510, 24.29520815942851807608926270357, 24.863594528431841422221852571005, 26.081613285612851001020274978035, 26.44714715149205564145927884240, 28.027497761808979891662752390674, 28.64824366844807251741877257800, 29.70992808462503245421640162088

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