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Properties

Label 1-621-621.263-r0-0-0
Degree $1$
Conductor $621$
Sign $0.552 - 0.833i$
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  + (−0.527 − 0.849i)2-s + (−0.444 + 0.895i)4-s + (0.873 + 0.486i)5-s + (0.386 − 0.922i)7-s + (0.995 − 0.0950i)8-s + (−0.0475 − 0.998i)10-s + (−0.204 + 0.978i)11-s + (0.991 + 0.126i)13-s + (−0.987 + 0.158i)14-s + (−0.605 − 0.795i)16-s + (0.235 − 0.971i)17-s + (−0.235 − 0.971i)19-s + (−0.823 + 0.567i)20-s + (0.939 − 0.342i)22-s + (0.527 + 0.849i)25-s + (−0.415 − 0.909i)26-s + ⋯
L(s)  = 1  + (−0.527 − 0.849i)2-s + (−0.444 + 0.895i)4-s + (0.873 + 0.486i)5-s + (0.386 − 0.922i)7-s + (0.995 − 0.0950i)8-s + (−0.0475 − 0.998i)10-s + (−0.204 + 0.978i)11-s + (0.991 + 0.126i)13-s + (−0.987 + 0.158i)14-s + (−0.605 − 0.795i)16-s + (0.235 − 0.971i)17-s + (−0.235 − 0.971i)19-s + (−0.823 + 0.567i)20-s + (0.939 − 0.342i)22-s + (0.527 + 0.849i)25-s + (−0.415 − 0.909i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $0.552 - 0.833i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ 0.552 - 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.161953802 - 0.6236400216i\)
\(L(\frac12)\) \(\approx\) \(1.161953802 - 0.6236400216i\)
\(L(1)\) \(\approx\) \(0.9562984686 - 0.3455337816i\)
\(L(1)\) \(\approx\) \(0.9562984686 - 0.3455337816i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.527 - 0.849i)T \)
5 \( 1 + (0.873 + 0.486i)T \)
7 \( 1 + (0.386 - 0.922i)T \)
11 \( 1 + (-0.204 + 0.978i)T \)
13 \( 1 + (0.991 + 0.126i)T \)
17 \( 1 + (0.235 - 0.971i)T \)
19 \( 1 + (-0.235 - 0.971i)T \)
29 \( 1 + (0.553 - 0.832i)T \)
31 \( 1 + (-0.823 - 0.567i)T \)
37 \( 1 + (0.327 + 0.945i)T \)
41 \( 1 + (0.0158 + 0.999i)T \)
43 \( 1 + (0.0792 + 0.996i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (0.605 - 0.795i)T \)
61 \( 1 + (-0.967 - 0.251i)T \)
67 \( 1 + (0.745 - 0.666i)T \)
71 \( 1 + (0.786 + 0.618i)T \)
73 \( 1 + (0.235 + 0.971i)T \)
79 \( 1 + (-0.296 + 0.954i)T \)
83 \( 1 + (-0.0158 + 0.999i)T \)
89 \( 1 + (0.580 - 0.814i)T \)
97 \( 1 + (-0.356 - 0.934i)T \)
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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

−23.47237938337100973625497798871, −22.28285559284827149181031388667, −21.451383488980391766220979103, −20.825112109549346201284808887640, −19.58953706386750166972570919086, −18.66425480186145433590404597136, −18.14089824836401295718212344859, −17.33588106541580813136296054191, −16.39596269006968319464763236566, −15.906237656540027782859788222029, −14.75143112963869710781289110608, −14.12417187391879201644530513867, −13.20781676706484671044809629318, −12.3171233323687959033250274056, −10.87516912341532297482890117112, −10.31134932061372831244646742347, −8.891266569659456126298692906400, −8.77611801168814722603054410279, −7.807376451073237894379524218888, −6.31311919871578779930782138687, −5.79409899414142927221811447778, −5.164554819463081793446387903486, −3.72039484518726632088631518354, −2.069274400063241099966387939192, −1.16211780176395265025792864191, 1.00089697559621231275634377176, 2.03992562057094828390375008223, 2.976994817622399238815333766345, 4.16765847556494722643882264565, 5.04730776906480779518007565795, 6.588078710192377824458857701914, 7.3509583978463779699614472651, 8.34685921484172182446584748661, 9.556352894470124371868691459852, 9.97545804087479479687825714528, 11.00070948569650256487350238102, 11.486518980772919245667230116113, 12.87326137008392153340336833036, 13.49387624349650029303277013983, 14.163518153867198411138481595644, 15.3565480952124907654636201043, 16.61385998580278437269956383359, 17.27588274236051937226949864036, 18.132826199272633117584023205466, 18.44141867476352989008053964541, 19.76478452924024021781595854238, 20.418993427852404074871522524034, 21.080189882313097977955400834583, 21.79090584845756804648186704055, 22.88028698336621932111013247040

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