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Properties

Label 1-621-621.193-r0-0-0
Degree $1$
Conductor $621$
Sign $0.664 + 0.747i$
Analytic cond. $2.88391$
Root an. cond. $2.88391$
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  + (0.991 + 0.126i)2-s + (0.967 + 0.251i)4-s + (0.997 + 0.0634i)5-s + (−0.857 + 0.513i)7-s + (0.928 + 0.371i)8-s + (0.981 + 0.189i)10-s + (−0.975 − 0.220i)11-s + (−0.0158 + 0.999i)13-s + (−0.916 + 0.400i)14-s + (0.873 + 0.486i)16-s + (0.580 − 0.814i)17-s + (0.580 + 0.814i)19-s + (0.950 + 0.312i)20-s + (−0.939 − 0.342i)22-s + (0.991 + 0.126i)25-s + (−0.142 + 0.989i)26-s + ⋯
L(s)  = 1  + (0.991 + 0.126i)2-s + (0.967 + 0.251i)4-s + (0.997 + 0.0634i)5-s + (−0.857 + 0.513i)7-s + (0.928 + 0.371i)8-s + (0.981 + 0.189i)10-s + (−0.975 − 0.220i)11-s + (−0.0158 + 0.999i)13-s + (−0.916 + 0.400i)14-s + (0.873 + 0.486i)16-s + (0.580 − 0.814i)17-s + (0.580 + 0.814i)19-s + (0.950 + 0.312i)20-s + (−0.939 − 0.342i)22-s + (0.991 + 0.126i)25-s + (−0.142 + 0.989i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $0.664 + 0.747i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ 0.664 + 0.747i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.635718017 + 1.183850659i\)
\(L(\frac12)\) \(\approx\) \(2.635718017 + 1.183850659i\)
\(L(1)\) \(\approx\) \(2.003294541 + 0.4682123450i\)
\(L(1)\) \(\approx\) \(2.003294541 + 0.4682123450i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.991 + 0.126i)T \)
5 \( 1 + (0.997 + 0.0634i)T \)
7 \( 1 + (-0.857 + 0.513i)T \)
11 \( 1 + (-0.975 - 0.220i)T \)
13 \( 1 + (-0.0158 + 0.999i)T \)
17 \( 1 + (0.580 - 0.814i)T \)
19 \( 1 + (0.580 + 0.814i)T \)
29 \( 1 + (-0.266 + 0.963i)T \)
31 \( 1 + (0.950 - 0.312i)T \)
37 \( 1 + (0.235 + 0.971i)T \)
41 \( 1 + (-0.553 + 0.832i)T \)
43 \( 1 + (-0.204 - 0.978i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (0.873 - 0.486i)T \)
61 \( 1 + (-0.999 - 0.0317i)T \)
67 \( 1 + (0.296 - 0.954i)T \)
71 \( 1 + (-0.888 - 0.458i)T \)
73 \( 1 + (0.580 + 0.814i)T \)
79 \( 1 + (-0.987 + 0.158i)T \)
83 \( 1 + (-0.553 - 0.832i)T \)
89 \( 1 + (-0.786 - 0.618i)T \)
97 \( 1 + (0.805 - 0.592i)T \)
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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

−22.79911065776984845489350773856, −22.15821333713119387941125765218, −21.247335241916642112543267487653, −20.650164119661013784805166393722, −19.83001909258810654541541791226, −18.9651064236525059320420355657, −17.76321101977953258897430373993, −17.02416814504292821913773354955, −15.998718978502356352052215439724, −15.36665253230289001391130384728, −14.32044639736514696118397375501, −13.40842819271681216662668982546, −13.03314896754728340243360295619, −12.26036861509136174614969518539, −10.84204694665284424381374699764, −10.26661808731401070036330985847, −9.55935974913729079748276134427, −7.97901648061161145808284368362, −7.060144540566166808794815546591, −6.02048343060964942275364566287, −5.47223940685356776281968839345, −4.38918408441786919444344299147, −3.12854937582456287558164288797, −2.522223893970527059997891803148, −1.103231109493541022519467719678, 1.654025713266938727843751377730, 2.69967988939440105486838258232, 3.38409465041006376862878200387, 4.880676532795397733812353560040, 5.56100947524883578534584985092, 6.382147655598576866230241620728, 7.175536657050593350240833944197, 8.4305700603334879354120577065, 9.67218824537901610873736555935, 10.26140477210442963128804930270, 11.511087386762349951290029069247, 12.29291179492564353679524089768, 13.20220628155121590233948537597, 13.77143227897279471948237195205, 14.550946510176719069290994666843, 15.60599573657607085821540625724, 16.36475570694704599811672946299, 16.92886617127200355522819693153, 18.39336976814283198205170212928, 18.80962477956670969974677046885, 20.141735081227660278474111969752, 20.91979176131418896490924833783, 21.549397345872871897716752798709, 22.24931205099885424981517427220, 22.98914597841279452746491383693

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