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Properties

Label 1-547-547.139-r0-0-0
Degree $1$
Conductor $547$
Sign $0.960 + 0.279i$
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  + (−0.910 − 0.413i)2-s + (−0.900 − 0.433i)3-s + (0.658 + 0.752i)4-s + (−0.684 − 0.729i)5-s + (0.641 + 0.767i)6-s + (−0.991 + 0.126i)7-s + (−0.289 − 0.957i)8-s + (0.623 + 0.781i)9-s + (0.322 + 0.946i)10-s + (−0.0402 − 0.999i)11-s + (−0.267 − 0.963i)12-s + (−0.733 + 0.680i)13-s + (0.955 + 0.294i)14-s + (0.300 + 0.953i)15-s + (−0.131 + 0.991i)16-s + (−0.109 − 0.994i)17-s + ⋯
L(s)  = 1  + (−0.910 − 0.413i)2-s + (−0.900 − 0.433i)3-s + (0.658 + 0.752i)4-s + (−0.684 − 0.729i)5-s + (0.641 + 0.767i)6-s + (−0.991 + 0.126i)7-s + (−0.289 − 0.957i)8-s + (0.623 + 0.781i)9-s + (0.322 + 0.946i)10-s + (−0.0402 − 0.999i)11-s + (−0.267 − 0.963i)12-s + (−0.733 + 0.680i)13-s + (0.955 + 0.294i)14-s + (0.300 + 0.953i)15-s + (−0.131 + 0.991i)16-s + (−0.109 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.960 + 0.279i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.960 + 0.279i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1711220443 + 0.02438334364i\)
\(L(\frac12)\) \(\approx\) \(0.1711220443 + 0.02438334364i\)
\(L(1)\) \(\approx\) \(0.3177443061 - 0.1329865763i\)
\(L(1)\) \(\approx\) \(0.3177443061 - 0.1329865763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.910 - 0.413i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.684 - 0.729i)T \)
7 \( 1 + (-0.991 + 0.126i)T \)
11 \( 1 + (-0.0402 - 0.999i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
17 \( 1 + (-0.109 - 0.994i)T \)
19 \( 1 + (-0.857 - 0.514i)T \)
23 \( 1 + (-0.614 + 0.788i)T \)
29 \( 1 + (-0.952 - 0.305i)T \)
31 \( 1 + (0.978 + 0.205i)T \)
37 \( 1 + (-0.999 - 0.0115i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.459 + 0.888i)T \)
47 \( 1 + (0.987 + 0.160i)T \)
53 \( 1 + (-0.976 - 0.216i)T \)
59 \( 1 + (0.692 + 0.721i)T \)
61 \( 1 + (-0.890 + 0.454i)T \)
67 \( 1 + (0.968 + 0.250i)T \)
71 \( 1 + (-0.577 + 0.816i)T \)
73 \( 1 + (0.300 - 0.953i)T \)
79 \( 1 + (-0.539 - 0.842i)T \)
83 \( 1 + (0.428 - 0.903i)T \)
89 \( 1 + (0.0517 + 0.998i)T \)
97 \( 1 + (0.983 - 0.183i)T \)
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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

−23.2968844046951174518396811894, −22.65799124008275445237091838192, −21.98682704914712960326989885942, −20.57910407964728231345921510226, −19.82582939345361759020489560708, −18.957271538203443056265603382337, −18.29584859519208043544395073946, −17.21931053055159697537964987339, −16.844208895981632025710813262619, −15.61832865127841791397545477010, −15.33236879226527690914801901782, −14.48739785140475674598082578551, −12.65021392273931662092912303590, −12.11052498286370982321300137729, −10.89894077254042443187506360195, −10.21906045181053631239970495935, −9.83071789839726726196274206812, −8.42772308044974864304280194162, −7.3525344462422655421316333924, −6.63457347360584275025929375560, −5.93918766228270918195902211684, −4.624678743691760658860814088643, −3.49151542833171533565737961421, −2.08957323968639407685273033619, −0.21922189110497038422410189122, 0.69466172231672250959630307717, 2.092819101783581135817903911856, 3.38782897531409770932459550468, 4.56887286049567452651382803683, 5.85876438635839121613047859247, 6.85948242024364259691968541902, 7.583905154699065554295735784, 8.70228269239218808079074899439, 9.48092313930217774860867187832, 10.50311413610064962621292478766, 11.594416999219919488214706714485, 11.90986100648360487273623017275, 12.862276191678307171704513473384, 13.60143897653424969629332729307, 15.58826069732543398027309379379, 16.06316802854872942300199634187, 16.815387142103628469989619464383, 17.37299881127184914211416431808, 18.62185771051712895464722438710, 19.213608384023653272562971602008, 19.63596266697025278765172965820, 20.838755017243892839432758722328, 21.76819322449423173126442447983, 22.45036701757513813119434347374, 23.556109595112602136778152015840

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