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Properties

Label 1-547-547.306-r0-0-0
Degree $1$
Conductor $547$
Sign $0.918 + 0.394i$
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  + (0.586 − 0.809i)2-s + (−0.900 − 0.433i)3-s + (−0.311 − 0.950i)4-s + (0.740 + 0.671i)5-s + (−0.880 + 0.474i)6-s + (−0.980 − 0.194i)7-s + (−0.952 − 0.305i)8-s + (0.623 + 0.781i)9-s + (0.978 − 0.205i)10-s + (−0.996 + 0.0804i)11-s + (−0.131 + 0.991i)12-s + (0.955 + 0.294i)13-s + (−0.733 + 0.680i)14-s + (−0.376 − 0.926i)15-s + (−0.806 + 0.591i)16-s + (−0.439 + 0.898i)17-s + ⋯
L(s)  = 1  + (0.586 − 0.809i)2-s + (−0.900 − 0.433i)3-s + (−0.311 − 0.950i)4-s + (0.740 + 0.671i)5-s + (−0.880 + 0.474i)6-s + (−0.980 − 0.194i)7-s + (−0.952 − 0.305i)8-s + (0.623 + 0.781i)9-s + (0.978 − 0.205i)10-s + (−0.996 + 0.0804i)11-s + (−0.131 + 0.991i)12-s + (0.955 + 0.294i)13-s + (−0.733 + 0.680i)14-s + (−0.376 − 0.926i)15-s + (−0.806 + 0.591i)16-s + (−0.439 + 0.898i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.918 + 0.394i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (306, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.918 + 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8139957742 + 0.1673856136i\)
\(L(\frac12)\) \(\approx\) \(0.8139957742 + 0.1673856136i\)
\(L(1)\) \(\approx\) \(0.8504973054 - 0.2815148443i\)
\(L(1)\) \(\approx\) \(0.8504973054 - 0.2815148443i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.586 - 0.809i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (0.740 + 0.671i)T \)
7 \( 1 + (-0.980 - 0.194i)T \)
11 \( 1 + (-0.996 + 0.0804i)T \)
13 \( 1 + (0.955 + 0.294i)T \)
17 \( 1 + (-0.439 + 0.898i)T \)
19 \( 1 + (-0.965 + 0.261i)T \)
23 \( 1 + (0.999 - 0.0230i)T \)
29 \( 1 + (0.0517 + 0.998i)T \)
31 \( 1 + (-0.999 + 0.0345i)T \)
37 \( 1 + (0.641 - 0.767i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.166 + 0.986i)T \)
47 \( 1 + (0.948 + 0.316i)T \)
53 \( 1 + (-0.614 + 0.788i)T \)
59 \( 1 + (-0.0402 + 0.999i)T \)
61 \( 1 + (-0.267 - 0.963i)T \)
67 \( 1 + (0.924 - 0.381i)T \)
71 \( 1 + (-0.944 - 0.327i)T \)
73 \( 1 + (-0.376 + 0.926i)T \)
79 \( 1 + (0.770 - 0.636i)T \)
83 \( 1 + (-0.632 - 0.774i)T \)
89 \( 1 + (-0.700 - 0.713i)T \)
97 \( 1 + (0.143 + 0.989i)T \)
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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

−23.441501635639076607222692552479, −22.492487440130202328153017509679, −21.92225778092805847021043287500, −20.96259984605577904216712655993, −20.569323182548724540046300378732, −18.748758638602230332028738537107, −18.019687000680599609121732239277, −17.12205733964458749301193189515, −16.518197669536163285045497492431, −15.71318946500083834459475720438, −15.28850069537235009723309364266, −13.696047048956980288465140332808, −13.06298522070802209206983717044, −12.569806228517560875672999900849, −11.36832679407578017452657567033, −10.30202368799588485939641971168, −9.28668960165566325796357840343, −8.57586693564210553505895002958, −7.098831737360972901647122839254, −6.24351514167236100343048924401, −5.567212417704462837543418634465, −4.85297101299407957145934131681, −3.7694581627819673885157186500, −2.53756776084448616672647932202, −0.40737317121052505986697702312, 1.36055475767389529278957483789, 2.38337253270919211618651078840, 3.444037082380410137768093938775, 4.646808539543878500338751402111, 5.880568464972652418074082800611, 6.22084041325869256677193041174, 7.20048243920488429802260452500, 8.89873092142236543812059342793, 10.003443955111803098209490712710, 10.848898557594770707190153957595, 11.00277457545420985813527491530, 12.6995134454381375600182940582, 12.89804500737337886511638304426, 13.65347611413932162426296214620, 14.78892483527132808656715947116, 15.7758623531708970063227106303, 16.77314931961985668496371243868, 17.83094504130934884518104487863, 18.59139700793680656290223470706, 19.02517066940401385751692650548, 20.11261234413580204806893783285, 21.37018032565352052216188033028, 21.67422060803152793289965735216, 22.633172590335218281909454528152, 23.351945427266950615106135174738

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