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Properties

Label 1-547-547.12-r1-0-0
Degree $1$
Conductor $547$
Sign $-0.930 + 0.365i$
Analytic cond. $58.7833$
Root an. cond. $58.7833$
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  + (0.244 − 0.969i)2-s + (−0.623 − 0.781i)3-s + (−0.880 − 0.474i)4-s + (−0.300 + 0.953i)5-s + (−0.910 + 0.413i)6-s + (−0.343 + 0.939i)7-s + (−0.675 + 0.736i)8-s + (−0.222 + 0.974i)9-s + (0.851 + 0.524i)10-s + (−0.0402 + 0.999i)11-s + (0.177 + 0.984i)12-s + (0.0747 − 0.997i)13-s + (0.826 + 0.563i)14-s + (0.932 − 0.359i)15-s + (0.548 + 0.835i)16-s + (0.332 + 0.942i)17-s + ⋯
L(s)  = 1  + (0.244 − 0.969i)2-s + (−0.623 − 0.781i)3-s + (−0.880 − 0.474i)4-s + (−0.300 + 0.953i)5-s + (−0.910 + 0.413i)6-s + (−0.343 + 0.939i)7-s + (−0.675 + 0.736i)8-s + (−0.222 + 0.974i)9-s + (0.851 + 0.524i)10-s + (−0.0402 + 0.999i)11-s + (0.177 + 0.984i)12-s + (0.0747 − 0.997i)13-s + (0.826 + 0.563i)14-s + (0.932 − 0.359i)15-s + (0.548 + 0.835i)16-s + (0.332 + 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.930 + 0.365i$
Analytic conductor: \(58.7833\)
Root analytic conductor: \(58.7833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (1:\ ),\ -0.930 + 0.365i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01101787388 + 0.05815424201i\)
\(L(\frac12)\) \(\approx\) \(0.01101787388 + 0.05815424201i\)
\(L(1)\) \(\approx\) \(0.6336827019 - 0.2013032311i\)
\(L(1)\) \(\approx\) \(0.6336827019 - 0.2013032311i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.244 - 0.969i)T \)
3 \( 1 + (-0.623 - 0.781i)T \)
5 \( 1 + (-0.300 + 0.953i)T \)
7 \( 1 + (-0.343 + 0.939i)T \)
11 \( 1 + (-0.0402 + 0.999i)T \)
13 \( 1 + (0.0747 - 0.997i)T \)
17 \( 1 + (0.332 + 0.942i)T \)
19 \( 1 + (-0.937 - 0.349i)T \)
23 \( 1 + (-0.233 + 0.972i)T \)
29 \( 1 + (0.990 + 0.137i)T \)
31 \( 1 + (0.418 + 0.908i)T \)
37 \( 1 + (-0.895 + 0.444i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.968 + 0.250i)T \)
47 \( 1 + (0.987 - 0.160i)T \)
53 \( 1 + (-0.778 - 0.627i)T \)
59 \( 1 + (-0.692 + 0.721i)T \)
61 \( 1 + (-0.999 - 0.0230i)T \)
67 \( 1 + (-0.763 + 0.645i)T \)
71 \( 1 + (-0.874 - 0.484i)T \)
73 \( 1 + (0.932 + 0.359i)T \)
79 \( 1 + (0.700 + 0.713i)T \)
83 \( 1 + (-0.428 - 0.903i)T \)
89 \( 1 + (0.479 - 0.877i)T \)
97 \( 1 + (0.469 + 0.882i)T \)
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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

−23.1042102357173920148745012985, −22.12162316795576306422319640319, −21.19465826393919536612853643137, −20.679678673918870264533152457051, −19.398058973361861727797071055107, −18.41447202111691243596027334671, −17.132714347083676502211417310182, −16.715042894590285034046481574722, −16.23206850478724571679030181655, −15.51394020945419799339657566537, −14.25740715932082184712480807910, −13.6517951658967098573510657416, −12.53650540765123894379574136682, −11.75344084484413385754991413438, −10.60155663439385801660964983961, −9.54042476751844658090656676530, −8.81335073014642988549653739454, −7.87305205814154903190393663752, −6.618226046883126717531311460852, −5.95801876754098499107003595377, −4.71793276470359886967909481783, −4.30879970360300261876740885253, −3.33498479787527365283666473537, −0.80562288008769873496226805656, −0.020759589446220303528737408897, 1.58470286139350290473031922437, 2.489475106965537336397138440326, 3.37930945657421350032378239101, 4.78128579394844129441369602883, 5.78908244571534513990235284501, 6.55818586065848018680654595068, 7.767060676159152754322778142566, 8.73309371383277516038416060078, 10.21122950016061615374236700368, 10.55727169326770309468926310051, 11.71823906952047980539914532018, 12.31375786183563813947650604169, 12.92118584813323069151715791328, 13.964179092726510220632598392, 15.0302500984321943373394623056, 15.59479786846344139263761674398, 17.40924886278697880967982549677, 17.80257028488308424751283092866, 18.6333957241725317918783594239, 19.37496233037826004572847046672, 19.82916666887197829582058618984, 21.30951798451536412506115616699, 21.954560773226137947393360166715, 22.74385966129324901293347185211, 23.19946774050723539333458147587

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