Properties

Label 1-547-547.217-r0-0-0
Degree $1$
Conductor $547$
Sign $0.555 - 0.831i$
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  + (−0.0402 − 0.999i)2-s + 3-s + (−0.996 + 0.0804i)4-s + (0.799 + 0.600i)5-s + (−0.0402 − 0.999i)6-s + (−0.845 − 0.534i)7-s + (0.120 + 0.992i)8-s + 9-s + (0.568 − 0.822i)10-s + (0.948 + 0.316i)11-s + (−0.996 + 0.0804i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (0.799 + 0.600i)15-s + (0.987 − 0.160i)16-s + (−0.200 + 0.979i)17-s + ⋯
L(s)  = 1  + (−0.0402 − 0.999i)2-s + 3-s + (−0.996 + 0.0804i)4-s + (0.799 + 0.600i)5-s + (−0.0402 − 0.999i)6-s + (−0.845 − 0.534i)7-s + (0.120 + 0.992i)8-s + 9-s + (0.568 − 0.822i)10-s + (0.948 + 0.316i)11-s + (−0.996 + 0.0804i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (0.799 + 0.600i)15-s + (0.987 − 0.160i)16-s + (−0.200 + 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.555 - 0.831i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.727671593 - 0.9235861573i\)
\(L(\frac12)\) \(\approx\) \(1.727671593 - 0.9235861573i\)
\(L(1)\) \(\approx\) \(1.350989850 - 0.5650427063i\)
\(L(1)\) \(\approx\) \(1.350989850 - 0.5650427063i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.0402 - 0.999i)T \)
3 \( 1 + T \)
5 \( 1 + (0.799 + 0.600i)T \)
7 \( 1 + (-0.845 - 0.534i)T \)
11 \( 1 + (0.948 + 0.316i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.200 + 0.979i)T \)
19 \( 1 + (0.987 + 0.160i)T \)
23 \( 1 + (0.692 - 0.721i)T \)
29 \( 1 + (-0.970 + 0.239i)T \)
31 \( 1 + (-0.354 + 0.935i)T \)
37 \( 1 + (0.692 + 0.721i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.428 - 0.903i)T \)
47 \( 1 + (0.278 - 0.960i)T \)
53 \( 1 + (-0.919 + 0.391i)T \)
59 \( 1 + (0.987 - 0.160i)T \)
61 \( 1 + (-0.0402 - 0.999i)T \)
67 \( 1 + (0.428 - 0.903i)T \)
71 \( 1 + (-0.632 + 0.774i)T \)
73 \( 1 + (0.799 - 0.600i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (-0.919 + 0.391i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (0.948 - 0.316i)T \)
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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

−23.96824816715615995572056817345, −22.50947710455592338652442467322, −21.98099561321260346864424995388, −21.14971834070156208102138933876, −20.03675654469774278298134645406, −19.20329506796309932244588904093, −18.50503546793739648681402875545, −17.520157136321390641185705430741, −16.45047698579738855892636749586, −16.08784353826625577245637921054, −14.91844097303527864203584590453, −14.23144510932650765238929084794, −13.42440225677955622655829265914, −12.90103832904575530238261239351, −11.67996423419095266208800997677, −9.64840251727133203943764186615, −9.42206332131731987003109619545, −8.961860742801152450842774482877, −7.65375136494775491945078402503, −6.80861996026600606137412689433, −5.89216510409205776810071824499, −4.844809941620562257541204927890, −3.793605082176175852610119152654, −2.60862027205079652321239117357, −1.21334432167934715828940422919, 1.25145771696139404023007147918, 2.28192425374310774513787150972, 3.26320297584587117926703658720, 3.81301413846594706211820260482, 5.20585549430350456191754305557, 6.58925775468659036636645571338, 7.50340190715446255011885691464, 8.77621798965665519799840841123, 9.54283789526673021683128460747, 10.15005311091221960028723994505, 10.85166635103858108348552299426, 12.37949732217136685319762848816, 12.96804367898957161763739193207, 13.79766104119114260872654418494, 14.47314048219857276571230698193, 15.244898427161549106035357815485, 16.803330145077316757807486179301, 17.520167426976386301002496657, 18.56337703447632872192857449527, 19.16665879900602322535114524107, 20.155388012022255822830026852401, 20.35274107664159264158476529686, 21.62782132760631377214287305891, 22.22385168761776503587007279397, 22.79379260582056147084311163682

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