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Properties

Label 1-547-547.6-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.205 - 0.978i$
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  + (0.233 − 0.972i)2-s + (0.623 − 0.781i)3-s + (−0.890 − 0.454i)4-s + (0.983 − 0.183i)5-s + (−0.614 − 0.788i)6-s + (0.740 + 0.671i)7-s + (−0.650 + 0.759i)8-s + (−0.222 − 0.974i)9-s + (0.0517 − 0.998i)10-s + (0.692 + 0.721i)11-s + (−0.910 + 0.413i)12-s + (0.0747 + 0.997i)13-s + (0.826 − 0.563i)14-s + (0.469 − 0.882i)15-s + (0.586 + 0.809i)16-s + (0.874 − 0.484i)17-s + ⋯
L(s)  = 1  + (0.233 − 0.972i)2-s + (0.623 − 0.781i)3-s + (−0.890 − 0.454i)4-s + (0.983 − 0.183i)5-s + (−0.614 − 0.788i)6-s + (0.740 + 0.671i)7-s + (−0.650 + 0.759i)8-s + (−0.222 − 0.974i)9-s + (0.0517 − 0.998i)10-s + (0.692 + 0.721i)11-s + (−0.910 + 0.413i)12-s + (0.0747 + 0.997i)13-s + (0.826 − 0.563i)14-s + (0.469 − 0.882i)15-s + (0.586 + 0.809i)16-s + (0.874 − 0.484i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.205 - 0.978i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.205 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.463657671 - 1.803009177i\)
\(L(\frac12)\) \(\approx\) \(1.463657671 - 1.803009177i\)
\(L(1)\) \(\approx\) \(1.325591748 - 1.045539979i\)
\(L(1)\) \(\approx\) \(1.325591748 - 1.045539979i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.233 - 0.972i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (0.983 - 0.183i)T \)
7 \( 1 + (0.740 + 0.671i)T \)
11 \( 1 + (0.692 + 0.721i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
17 \( 1 + (0.874 - 0.484i)T \)
19 \( 1 + (0.658 + 0.752i)T \)
23 \( 1 + (0.00575 - 0.999i)T \)
29 \( 1 + (-0.928 + 0.370i)T \)
31 \( 1 + (-0.700 - 0.713i)T \)
37 \( 1 + (-0.976 - 0.216i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.343 + 0.939i)T \)
47 \( 1 + (-0.996 + 0.0804i)T \)
53 \( 1 + (0.529 + 0.848i)T \)
59 \( 1 + (-0.919 + 0.391i)T \)
61 \( 1 + (0.895 + 0.444i)T \)
67 \( 1 + (0.0976 - 0.995i)T \)
71 \( 1 + (-0.763 - 0.645i)T \)
73 \( 1 + (0.469 + 0.882i)T \)
79 \( 1 + (-0.985 - 0.171i)T \)
83 \( 1 + (-0.845 - 0.534i)T \)
89 \( 1 + (-0.832 - 0.553i)T \)
97 \( 1 + (-0.937 + 0.349i)T \)
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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

−23.779809386968511843983630892094, −22.64416297652972911447163565870, −21.94516464038387316218379044408, −21.31859047258048836560937996718, −20.53234413308752240696966114883, −19.48222783516532583140143017820, −18.320239437727221918966859778611, −17.36515741458965395755260345474, −16.93011866317985169837504481530, −15.95313618976782253888931809062, −14.95736226160348467239183803018, −14.363357557377113398060326056778, −13.70469856374312579114735150121, −13.06416660316899314308166685068, −11.4258668657455013136701290587, −10.360070065434045842619071648333, −9.60439315628274395564690315977, −8.701662432905755433999333380921, −7.88857862764199607746719875721, −6.93849371292003892556905124831, −5.54383241529113963756459907099, −5.199848929837373969275851386737, −3.80033057521529760169375253204, −3.15786931720350093317573167455, −1.41079808971322420294231079296, 1.4642357492481477753266584531, 1.80992412732322669731012861315, 2.83008292661803523543664506479, 4.08412684902452750618418464519, 5.25471932392963858424823505132, 6.12352470360596332196336316560, 7.37231503418383069046625551069, 8.64936548856229351300797814372, 9.240719382270406207849061867575, 9.95412097629001938278684653573, 11.347224373669911134920329491418, 12.171632706601063449711996788720, 12.66785126540629578973782631563, 13.85233140763545907839264046082, 14.32888994178676400760578090643, 14.87905498924383828193277687537, 16.699289376125648906504141260425, 17.62682751478633674762871388031, 18.505589749404071333781580336610, 18.7073246602876179364610804201, 19.98965542023903933596412985474, 20.75902246280544857770115504611, 21.12222028898475473985965935962, 22.202916815218142244760184176172, 22.96514579946742881572801811831

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