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Properties

Label 1-547-547.489-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.224 + 0.974i$
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  + (0.863 + 0.504i)2-s + (0.623 + 0.781i)3-s + (0.490 + 0.871i)4-s + (0.874 + 0.484i)5-s + (0.143 + 0.989i)6-s + (−0.439 − 0.898i)7-s + (−0.0172 + 0.999i)8-s + (−0.222 + 0.974i)9-s + (0.509 + 0.860i)10-s + (0.799 − 0.600i)11-s + (−0.376 + 0.926i)12-s + (0.826 + 0.563i)13-s + (0.0747 − 0.997i)14-s + (0.166 + 0.986i)15-s + (−0.519 + 0.854i)16-s + (−0.177 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.863 + 0.504i)2-s + (0.623 + 0.781i)3-s + (0.490 + 0.871i)4-s + (0.874 + 0.484i)5-s + (0.143 + 0.989i)6-s + (−0.439 − 0.898i)7-s + (−0.0172 + 0.999i)8-s + (−0.222 + 0.974i)9-s + (0.509 + 0.860i)10-s + (0.799 − 0.600i)11-s + (−0.376 + 0.926i)12-s + (0.826 + 0.563i)13-s + (0.0747 − 0.997i)14-s + (0.166 + 0.986i)15-s + (−0.519 + 0.854i)16-s + (−0.177 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.224 + 0.974i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.224 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.994084316 + 2.506397662i\)
\(L(\frac12)\) \(\approx\) \(1.994084316 + 2.506397662i\)
\(L(1)\) \(\approx\) \(1.855020301 + 1.305511733i\)
\(L(1)\) \(\approx\) \(1.855020301 + 1.305511733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.863 + 0.504i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.874 + 0.484i)T \)
7 \( 1 + (-0.439 - 0.898i)T \)
11 \( 1 + (0.799 - 0.600i)T \)
13 \( 1 + (0.826 + 0.563i)T \)
17 \( 1 + (-0.177 - 0.984i)T \)
19 \( 1 + (-0.717 + 0.696i)T \)
23 \( 1 + (-0.397 - 0.917i)T \)
29 \( 1 + (0.256 - 0.966i)T \)
31 \( 1 + (-0.985 - 0.171i)T \)
37 \( 1 + (0.983 + 0.183i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.211 + 0.977i)T \)
47 \( 1 + (-0.845 + 0.534i)T \)
53 \( 1 + (-0.937 - 0.349i)T \)
59 \( 1 + (0.948 + 0.316i)T \)
61 \( 1 + (0.300 - 0.953i)T \)
67 \( 1 + (-0.614 - 0.788i)T \)
71 \( 1 + (-0.910 - 0.413i)T \)
73 \( 1 + (0.166 - 0.986i)T \)
79 \( 1 + (-0.952 - 0.305i)T \)
83 \( 1 + (0.692 + 0.721i)T \)
89 \( 1 + (0.675 - 0.736i)T \)
97 \( 1 + (-0.980 - 0.194i)T \)
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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

−23.281110418856445647497680901852, −21.96287041449596184260616615640, −21.6861674175776912382545177861, −20.541463046658462715011472817776, −19.92682674877392985692560175003, −19.22951807290780020483987947755, −18.20474843523095951921309592756, −17.53898005004698949889447682716, −16.164025947997129987699007651145, −15.074562051139179821031651852552, −14.581152343493406690110246797376, −13.42862417745909721446066120177, −12.93974948554915648594440212511, −12.380696422365134956133985947910, −11.36140879332674316445436378937, −10.07882025573986060458532078444, −9.19394639279713420926311257082, −8.516116074396489709598931731347, −6.90132443255638297673144952942, −6.169932427896916089897712568363, −5.47147828278704510476441400500, −4.0647139139453234000512320858, −3.034093815543520152248228124312, −2.00204789580701199788467647218, −1.36465232269173656774537417196, 1.93090647966439006588266438055, 3.075631581972110552437164445819, 3.85368302112684420290175665653, 4.6483248895202689865197361603, 6.04176640577995990397235830090, 6.54811373440220875948413631260, 7.73396201144930119031436472822, 8.81493842956607516780235896595, 9.69671116799542407134889851924, 10.73688613783864547170200075962, 11.43634102290707848381064429103, 12.97607600657115576861032555363, 13.717183519235106212792618704451, 14.21205225216769801034378069530, 14.833702061849061564112339781896, 16.18625787055776628237002714545, 16.46520117748931714748164693433, 17.357593944316545813348229665748, 18.63943626855815215082002674528, 19.69532725221070395555021672062, 20.741442338453946578881783651055, 21.05180163563306137826623628648, 22.1753146424329356742920314583, 22.499973927423258965712073519000, 23.481140047588478749538941941832

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