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Properties

Label 1-101-101.94-r1-0-0
Degree $1$
Conductor $101$
Sign $-0.956 - 0.291i$
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  + (−0.844 − 0.535i)2-s + (−0.248 − 0.968i)3-s + (0.425 + 0.904i)4-s + (0.535 + 0.844i)5-s + (−0.309 + 0.951i)6-s + (−0.368 + 0.929i)7-s + (0.125 − 0.992i)8-s + (−0.876 + 0.481i)9-s − i·10-s + (−0.481 − 0.876i)11-s + (0.770 − 0.637i)12-s + (0.929 − 0.368i)13-s + (0.809 − 0.587i)14-s + (0.684 − 0.728i)15-s + (−0.637 + 0.770i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.844 − 0.535i)2-s + (−0.248 − 0.968i)3-s + (0.425 + 0.904i)4-s + (0.535 + 0.844i)5-s + (−0.309 + 0.951i)6-s + (−0.368 + 0.929i)7-s + (0.125 − 0.992i)8-s + (−0.876 + 0.481i)9-s − i·10-s + (−0.481 − 0.876i)11-s + (0.770 − 0.637i)12-s + (0.929 − 0.368i)13-s + (0.809 − 0.587i)14-s + (0.684 − 0.728i)15-s + (−0.637 + 0.770i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.956 - 0.291i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ -0.956 - 0.291i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08030581240 - 0.5392848313i\)
\(L(\frac12)\) \(\approx\) \(0.08030581240 - 0.5392848313i\)
\(L(1)\) \(\approx\) \(0.5292342760 - 0.2811085605i\)
\(L(1)\) \(\approx\) \(0.5292342760 - 0.2811085605i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.844 - 0.535i)T \)
3 \( 1 + (-0.248 - 0.968i)T \)
5 \( 1 + (0.535 + 0.844i)T \)
7 \( 1 + (-0.368 + 0.929i)T \)
11 \( 1 + (-0.481 - 0.876i)T \)
13 \( 1 + (0.929 - 0.368i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.637 - 0.770i)T \)
23 \( 1 + (-0.0627 - 0.998i)T \)
29 \( 1 + (-0.368 - 0.929i)T \)
31 \( 1 + (-0.929 - 0.368i)T \)
37 \( 1 + (0.968 + 0.248i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + (0.187 - 0.982i)T \)
47 \( 1 + (0.187 + 0.982i)T \)
53 \( 1 + (-0.904 - 0.425i)T \)
59 \( 1 + (0.770 + 0.637i)T \)
61 \( 1 + (-0.904 + 0.425i)T \)
67 \( 1 + (0.248 - 0.968i)T \)
71 \( 1 + (0.968 - 0.248i)T \)
73 \( 1 + (0.998 - 0.0627i)T \)
79 \( 1 + (0.0627 - 0.998i)T \)
83 \( 1 + (-0.998 - 0.0627i)T \)
89 \( 1 + (-0.770 + 0.637i)T \)
97 \( 1 + (-0.425 - 0.904i)T \)
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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

−29.62063024183482259973321658305, −28.64576946822781387364277602859, −28.056383023128138509305792588755, −27.03645599854117920461431769558, −25.85784622855872600294794272321, −25.508294891057768622391842456083, −23.71224607805474340571640009060, −23.2503187546342673230515964358, −21.52123748160336861546783549970, −20.48209865370487182035212535595, −19.84453760142926517251772036842, −18.10967693879559163202935353091, −17.0887963228754057899920016475, −16.49378379768441551688510446800, −15.52596476240677146420020467112, −14.29615990180023607512919371213, −12.88778560700578362976047304203, −11.04760651440142665444550924597, −10.13643248074209836450580709478, −9.301395592897430390188422353687, −8.14983349315534599349203248359, −6.49798737795101469645369812627, −5.349826881783232305301816825001, −4.02886269041131843544147238267, −1.54064684616581696805904270206, 0.32023234531304627760826658075, 2.20125977038519455888702182448, 3.012172338416597457532756625433, 5.85902537215085152814131322261, 6.76041919690064613598880305486, 8.139845071533429593628834294941, 9.19747617812273692968233683739, 10.76651713865513919544647895278, 11.459137710432503944136921441198, 12.83077697549538741447803459891, 13.6608495825774142462788867277, 15.44595477030299204701617157528, 16.72702295907687488542492192130, 18.05090259392117806307546891796, 18.49640155998386930072467809722, 19.21065689861953168270929708404, 20.64898566553416300247352916382, 21.88451514227733155302877009257, 22.656925862181712137613007813252, 24.25546760913235575794554161570, 25.38240850211121046780871275463, 25.87941503471850874472640430425, 27.185204112186613650199402175558, 28.56950448501124751418004660181, 28.99786953058447688315505082501

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