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Properties

Label 1-101-101.12-r1-0-0
Degree $1$
Conductor $101$
Sign $0.941 - 0.338i$
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  + (−0.248 − 0.968i)2-s + (0.998 − 0.0627i)3-s + (−0.876 + 0.481i)4-s + (0.968 + 0.248i)5-s + (−0.309 − 0.951i)6-s + (−0.770 + 0.637i)7-s + (0.684 + 0.728i)8-s + (0.992 − 0.125i)9-s − i·10-s + (0.125 + 0.992i)11-s + (−0.844 + 0.535i)12-s + (0.637 − 0.770i)13-s + (0.809 + 0.587i)14-s + (0.982 + 0.187i)15-s + (0.535 − 0.844i)16-s + (−0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.248 − 0.968i)2-s + (0.998 − 0.0627i)3-s + (−0.876 + 0.481i)4-s + (0.968 + 0.248i)5-s + (−0.309 − 0.951i)6-s + (−0.770 + 0.637i)7-s + (0.684 + 0.728i)8-s + (0.992 − 0.125i)9-s − i·10-s + (0.125 + 0.992i)11-s + (−0.844 + 0.535i)12-s + (0.637 − 0.770i)13-s + (0.809 + 0.587i)14-s + (0.982 + 0.187i)15-s + (0.535 − 0.844i)16-s + (−0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.941 - 0.338i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ 0.941 - 0.338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.186603091 - 0.3808328061i\)
\(L(\frac12)\) \(\approx\) \(2.186603091 - 0.3808328061i\)
\(L(1)\) \(\approx\) \(1.385218825 - 0.3347323994i\)
\(L(1)\) \(\approx\) \(1.385218825 - 0.3347323994i\)

Euler product

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

−29.61687557508762285681697582208, −28.67468823942813440755938270815, −27.14467838457070272959466015348, −26.31499556936712346402179522033, −25.67551509252930833632307881213, −24.70853448788848825500727894095, −23.9032737345320894135823892027, −22.43151620817726600399765002510, −21.38092385105866321030380143216, −20.11777493166186491853042521819, −19.00179899842403478263870424496, −18.07576797699778557775573720215, −16.58542306831580347587030344136, −16.06781569500563465953402002296, −14.53553600394254757875568809823, −13.58473867974590899241863940226, −13.22647844370310390354532295914, −10.641343700594707473916020624856, −9.2558695573597922692515198183, −8.97283598880206116520764182343, −7.27692775836328209668074364216, −6.33393903740371489427062475277, −4.751379778348255063826232293747, −3.19392386489661366793618584608, −1.10698758404243245275019233545, 1.63358996489598996843178257955, 2.67805953636830819358146684288, 3.85467952967504005813715253833, 5.77563652042077724942802997262, 7.5635448292360776139089328271, 9.029992793909947598518998712599, 9.63897572030548299029768045010, 10.68396634034751229768782927096, 12.62046527389620790259492605796, 13.059356580527553359432602368212, 14.30503907427709895316283596219, 15.4466691353949306889131616044, 17.22482462357902887013154673146, 18.3129968434968762620815002352, 19.061173032792603524921266421052, 20.23652714825067635420144032332, 20.98433067663258269334960594997, 22.02184758035111454130519096983, 22.91640274702844549669299663065, 24.89381140205483514187741073739, 25.66706262497286727057739024356, 26.3049762459088049673300653374, 27.693971611305098448429116862335, 28.67110927578741301356621854730, 29.65205000565725117787888085612

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