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Properties

Label 1-201-201.137-r0-0-0
Degree $1$
Conductor $201$
Sign $-0.687 + 0.725i$
Analytic cond. $0.933440$
Root an. cond. $0.933440$
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  + (0.841 + 0.540i)2-s + (0.415 + 0.909i)4-s + (−0.654 + 0.755i)5-s + (−0.841 − 0.540i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)10-s + (−0.654 + 0.755i)11-s + (0.142 + 0.989i)13-s + (−0.415 − 0.909i)14-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + (0.841 − 0.540i)19-s + (−0.959 − 0.281i)20-s + (−0.959 + 0.281i)22-s + (0.959 + 0.281i)23-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (0.415 + 0.909i)4-s + (−0.654 + 0.755i)5-s + (−0.841 − 0.540i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)10-s + (−0.654 + 0.755i)11-s + (0.142 + 0.989i)13-s + (−0.415 − 0.909i)14-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + (0.841 − 0.540i)19-s + (−0.959 − 0.281i)20-s + (−0.959 + 0.281i)22-s + (0.959 + 0.281i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.687 + 0.725i$
Analytic conductor: \(0.933440\)
Root analytic conductor: \(0.933440\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 201,\ (0:\ ),\ -0.687 + 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5258615849 + 1.222942315i\)
\(L(\frac12)\) \(\approx\) \(0.5258615849 + 1.222942315i\)
\(L(1)\) \(\approx\) \(1.026372748 + 0.7656479065i\)
\(L(1)\) \(\approx\) \(1.026372748 + 0.7656479065i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.841 + 0.540i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (0.142 + 0.989i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 - T \)
31 \( 1 + (0.142 - 0.989i)T \)
37 \( 1 + T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (0.959 + 0.281i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 - T \)
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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

−26.7552553719916532284543286328, −25.08422716582236860336547551125, −24.66293352015286612652885278493, −23.45727070706359065874633886476, −22.78578150380048621109015587447, −21.87814062379648444458741050272, −20.72289635166183369960480044000, −20.11611071390770150261096896034, −19.097602589522889969584821922004, −18.29834448455303035724597625733, −16.401566510634703654893522127907, −15.85264206681340921931761620714, −14.963981686713909485150165698339, −13.45782949000260443809598550639, −12.89283413412035468400029451306, −11.94849711661368708873422016674, −10.9990055117642095610066344508, −9.76250639861285815040398301617, −8.65745990924392822759376007387, −7.247576129392121597160644562508, −5.755607166650076407420323975500, −5.05984178361529798230220819780, −3.575011137983380617559283541870, −2.75083019888413403885742086030, −0.776089820147681477182095972814, 2.47281155441932568263282125741, 3.636565022930741865750870959589, 4.508756878980769056623078816033, 6.069638926258411362776465027, 7.06771046111135330817956475979, 7.65685259591740330785558883197, 9.27395406406590113187570111861, 10.71687739704950325120953677333, 11.61823419549539959101517381332, 12.82399660785939857120476146181, 13.580464500606299281959339193217, 14.7790882872251403655662732206, 15.49466074673483640341355608395, 16.37824184853399008692804918075, 17.410919377479772291128873740127, 18.67989398753253127725972752290, 19.7281021890256342332879739802, 20.68616336074606880494853912481, 21.91393312383191510749315179413, 22.63556000188671269716031006208, 23.45222781450721613535598478301, 24.04752491668103125380909386814, 25.4766780646140932619354647762, 26.312808609629194403598523699405, 26.607004973984849105407403217962

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