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Properties

Label 1-621-621.169-r0-0-0
Degree $1$
Conductor $621$
Sign $0.920 - 0.391i$
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  + (−0.916 + 0.400i)2-s + (0.678 − 0.734i)4-s + (−0.204 − 0.978i)5-s + (−0.823 + 0.567i)7-s + (−0.327 + 0.945i)8-s + (0.580 + 0.814i)10-s + (0.997 + 0.0634i)11-s + (0.902 − 0.429i)13-s + (0.527 − 0.849i)14-s + (−0.0792 − 0.996i)16-s + (0.0475 + 0.998i)17-s + (0.0475 − 0.998i)19-s + (−0.857 − 0.513i)20-s + (−0.939 + 0.342i)22-s + (−0.916 + 0.400i)25-s + (−0.654 + 0.755i)26-s + ⋯
L(s)  = 1  + (−0.916 + 0.400i)2-s + (0.678 − 0.734i)4-s + (−0.204 − 0.978i)5-s + (−0.823 + 0.567i)7-s + (−0.327 + 0.945i)8-s + (0.580 + 0.814i)10-s + (0.997 + 0.0634i)11-s + (0.902 − 0.429i)13-s + (0.527 − 0.849i)14-s + (−0.0792 − 0.996i)16-s + (0.0475 + 0.998i)17-s + (0.0475 − 0.998i)19-s + (−0.857 − 0.513i)20-s + (−0.939 + 0.342i)22-s + (−0.916 + 0.400i)25-s + (−0.654 + 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ 0.920 - 0.391i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8129785905 - 0.1657303119i\)
\(L(\frac12)\) \(\approx\) \(0.8129785905 - 0.1657303119i\)
\(L(1)\) \(\approx\) \(0.7101860859 + 0.01454072553i\)
\(L(1)\) \(\approx\) \(0.7101860859 + 0.01454072553i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.916 + 0.400i)T \)
5 \( 1 + (-0.204 - 0.978i)T \)
7 \( 1 + (-0.823 + 0.567i)T \)
11 \( 1 + (0.997 + 0.0634i)T \)
13 \( 1 + (0.902 - 0.429i)T \)
17 \( 1 + (0.0475 + 0.998i)T \)
19 \( 1 + (0.0475 - 0.998i)T \)
29 \( 1 + (0.296 - 0.954i)T \)
31 \( 1 + (-0.857 + 0.513i)T \)
37 \( 1 + (0.928 + 0.371i)T \)
41 \( 1 + (-0.745 + 0.666i)T \)
43 \( 1 + (0.873 - 0.486i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.0792 + 0.996i)T \)
61 \( 1 + (0.630 - 0.776i)T \)
67 \( 1 + (-0.553 - 0.832i)T \)
71 \( 1 + (0.723 - 0.690i)T \)
73 \( 1 + (0.0475 - 0.998i)T \)
79 \( 1 + (-0.266 - 0.963i)T \)
83 \( 1 + (-0.745 - 0.666i)T \)
89 \( 1 + (0.981 + 0.189i)T \)
97 \( 1 + (0.472 - 0.881i)T \)
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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

−22.81461458215175511982781331726, −22.269839379142204210688821179206, −21.31624372052283410727120600614, −20.2763293358508997740963214889, −19.72268216243284365954741095704, −18.74364346152346723152438463961, −18.427042917813800578484384976285, −17.326481220778371578464150616946, −16.37554980866706551778208755688, −15.96934294586876646194536257754, −14.67330644719325450830165876400, −13.85954116847231315731527894364, −12.77369581313518638433820106341, −11.71295043978751739130842679584, −11.12587257617374401232317427608, −10.200439862990784142588249058455, −9.51150862222694457058365392586, −8.583785621494821437493748064246, −7.364447808304526818038190067659, −6.82760043316944484857519707639, −6.00255373064129368430187177721, −3.87051239788322378707766473735, −3.51441834604685824745831999326, −2.294264399725210136143491417784, −0.983791198567154861797855375701, 0.74862845774425849222759793016, 1.82561071195290561697081232718, 3.252761974274355703738688883047, 4.51634678100657168241951909388, 5.84394231967989357668523823187, 6.30070545124929215504850294983, 7.52714375781028858455870621356, 8.59697683721555014427240103493, 9.02204655051027087349202056725, 9.84031086168426485645683264025, 10.97150596161073148679284248091, 11.87981036373146616089360389998, 12.7353283251745729970058732968, 13.682071614684042195634855465313, 15.02843664311235852403499112417, 15.5905338793254724219711802334, 16.39289809013365788867810903146, 17.06734234037599419189293542250, 17.86266453381617571919483817349, 18.883449393569476377863301217298, 19.65450356631115463714901205046, 20.08585103935977516025911259996, 21.14385752518070496019791639575, 22.10563718564500977398346298691, 23.21375294794730678945975221521

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