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Properties

Label 1-101-101.72-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: $\chi_{101} (72, \cdot )$
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

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

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