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Properties

Label 1-547-547.468-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: $\chi_{547} (468, \cdot )$
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

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

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