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Properties

Label 1-547-547.90-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.115 - 0.993i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (0.813 − 0.582i)2-s + (−0.900 − 0.433i)3-s + (0.322 − 0.946i)4-s + (−0.289 + 0.957i)5-s + (−0.985 + 0.171i)6-s + (0.386 − 0.922i)7-s + (−0.289 − 0.957i)8-s + (0.623 + 0.781i)9-s + (0.322 + 0.946i)10-s + (0.885 + 0.464i)11-s + (−0.700 + 0.713i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.675 − 0.736i)15-s + (−0.792 − 0.609i)16-s + (0.915 + 0.402i)17-s + ⋯
L(s)  = 1  + (0.813 − 0.582i)2-s + (−0.900 − 0.433i)3-s + (0.322 − 0.946i)4-s + (−0.289 + 0.957i)5-s + (−0.985 + 0.171i)6-s + (0.386 − 0.922i)7-s + (−0.289 − 0.957i)8-s + (0.623 + 0.781i)9-s + (0.322 + 0.946i)10-s + (0.885 + 0.464i)11-s + (−0.700 + 0.713i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.675 − 0.736i)15-s + (−0.792 − 0.609i)16-s + (0.915 + 0.402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.115 - 0.993i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.115 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.123869493 - 1.261915028i\)
\(L(\frac12)\) \(\approx\) \(1.123869493 - 1.261915028i\)
\(L(1)\) \(\approx\) \(1.154183649 - 0.6589468705i\)
\(L(1)\) \(\approx\) \(1.154183649 - 0.6589468705i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.813 - 0.582i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.289 + 0.957i)T \)
7 \( 1 + (0.386 - 0.922i)T \)
11 \( 1 + (0.885 + 0.464i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.915 + 0.402i)T \)
19 \( 1 + (-0.0172 + 0.999i)T \)
23 \( 1 + (0.990 + 0.137i)T \)
29 \( 1 + (-0.952 - 0.305i)T \)
31 \( 1 + (0.978 + 0.205i)T \)
37 \( 1 + (0.509 - 0.860i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.539 - 0.842i)T \)
47 \( 1 + (-0.354 - 0.935i)T \)
53 \( 1 + (0.675 - 0.736i)T \)
59 \( 1 + (-0.970 + 0.239i)T \)
61 \( 1 + (0.0517 - 0.998i)T \)
67 \( 1 + (-0.700 + 0.713i)T \)
71 \( 1 + (-0.418 - 0.908i)T \)
73 \( 1 + (0.675 + 0.736i)T \)
79 \( 1 + (-0.539 - 0.842i)T \)
83 \( 1 + (0.568 + 0.822i)T \)
89 \( 1 + (0.0517 + 0.998i)T \)
97 \( 1 + (-0.650 - 0.759i)T \)
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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

−23.65542575598012388599648001166, −22.79473871601705198881239833734, −21.961801142799218940974831988423, −21.2953650299126740715954260589, −20.801861451850369154354060892123, −19.48373159818434310017627319960, −18.370443790773691040090180734895, −17.1945055196351085068176701626, −16.77533374967207574411606221611, −16.045664155753194265090718848008, −15.221204965342301762186943127090, −14.45774855602667200302410999371, −13.25486957643070161137287765295, −12.34755641360812088288722852382, −11.665607069503822642880949405910, −11.280155943925299713576356862686, −9.360758812137102299054687016970, −8.9034436649237806155632919733, −7.64219512300235115951805617445, −6.517321935227182913508720297812, −5.70812152325908816112941928281, −4.82439187835999713292281190743, −4.33021169890722045512937322574, −3.01689904222826525343024816251, −1.30935126393273553325104257376, 0.898023283474269676503605813938, 1.96124905292137217970715306343, 3.40001211064344713061634674151, 4.17423180402482165723715309433, 5.31986875653739312060981484439, 6.21468551668523411824949900263, 7.11124913037721092897020546684, 7.78495714986501973034733667866, 9.88693431876579259389348027315, 10.4496219545467303928016622414, 11.183887393126274855661261141067, 11.95942895019127879269011910831, 12.72418031088039320227482682241, 13.70473671860820167558067329249, 14.588469228652404213626748880250, 15.170696620983597690135641128076, 16.4879658555492628786727228745, 17.32915985720747548246212371701, 18.22265362514892568557355557027, 19.11474763961984314044616498524, 19.73335277003044432766641210640, 20.78471573409754086381443080971, 21.66313667627497703208288677599, 22.74014624822542094874957988846, 22.88589745051423163821193889577

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