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Properties

Label 1-101-101.66-r1-0-0
Degree $1$
Conductor $101$
Sign $-0.741 - 0.671i$
Analytic cond. $10.8539$
Root an. cond. $10.8539$
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  + (0.481 − 0.876i)2-s + (−0.125 + 0.992i)3-s + (−0.535 − 0.844i)4-s + (0.876 − 0.481i)5-s + (0.809 + 0.587i)6-s + (−0.982 + 0.187i)7-s + (−0.998 + 0.0627i)8-s + (−0.968 − 0.248i)9-s − i·10-s + (0.248 − 0.968i)11-s + (0.904 − 0.425i)12-s + (0.187 − 0.982i)13-s + (−0.309 + 0.951i)14-s + (0.368 + 0.929i)15-s + (−0.425 + 0.904i)16-s + (0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.481 − 0.876i)2-s + (−0.125 + 0.992i)3-s + (−0.535 − 0.844i)4-s + (0.876 − 0.481i)5-s + (0.809 + 0.587i)6-s + (−0.982 + 0.187i)7-s + (−0.998 + 0.0627i)8-s + (−0.968 − 0.248i)9-s − i·10-s + (0.248 − 0.968i)11-s + (0.904 − 0.425i)12-s + (0.187 − 0.982i)13-s + (−0.309 + 0.951i)14-s + (0.368 + 0.929i)15-s + (−0.425 + 0.904i)16-s + (0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.741 - 0.671i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ -0.741 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5255823441 - 1.363714688i\)
\(L(\frac12)\) \(\approx\) \(0.5255823441 - 1.363714688i\)
\(L(1)\) \(\approx\) \(0.9837559178 - 0.5852011162i\)
\(L(1)\) \(\approx\) \(0.9837559178 - 0.5852011162i\)

Euler product

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

−30.127449895359109352929855904358, −29.307320892700097772427713816229, −28.144934555222057295103295300775, −26.267961332155098586524085190522, −25.71007980237279060963160595560, −25.00864901957049780320576223335, −23.771578723662629254493931297966, −22.88936363563226178340338729527, −22.16078929141444363004866870588, −20.80497586450575742869900931480, −19.14735573225641504002668936669, −18.2628710180824916301363710787, −17.18422545119174979528760747435, −16.42733881848713609588041086099, −14.680527827354480011301120579896, −13.98478124024519927751680449220, −12.884947760601853988250396793464, −12.14132250864937776693716301891, −10.14941998045074180143992970372, −8.81577584776826033419265159708, −7.23703921322096642655365653400, −6.54428129872529361930125522831, −5.62572505409288237565961991219, −3.67588481065571054996948138302, −2.01166035506578871393531619432, 0.54017596935123967026672580138, 2.712935276915794715725652427858, 3.778589013730888860991656397139, 5.39268906945053583201919632448, 5.96983095779782026389607548757, 8.81738276095947520958994876402, 9.63935882483211659910200634945, 10.49130450282906831842631743838, 11.74880992959062388946321568576, 13.06540688081696853656858056480, 13.89721296873427824797805748438, 15.25685048734381904491108595249, 16.36162802553667885901493736065, 17.54570797565861695400528657530, 18.961700913907053992005962076696, 20.14376654938601811808704855891, 20.941851890893480246948778237154, 21.99233107890763966395811704790, 22.41942693938398633113339874993, 23.783220612713836201511984185853, 25.199341964133794080381838902568, 26.25271274105024341204756980510, 27.64845721441804771476864209385, 28.20973262641398985322404074560, 29.35625510779705781757754244438

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