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Properties

Label 1-547-547.74-r0-0-0
Degree $1$
Conductor $547$
Sign $0.000102 - 0.999i$
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  + (0.785 − 0.618i)2-s + (−0.900 + 0.433i)3-s + (0.233 − 0.972i)4-s + (0.995 − 0.0919i)5-s + (−0.439 + 0.898i)6-s + (0.932 + 0.359i)7-s + (−0.418 − 0.908i)8-s + (0.623 − 0.781i)9-s + (0.725 − 0.688i)10-s + (−0.919 − 0.391i)11-s + (0.211 + 0.977i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (−0.857 + 0.514i)15-s + (−0.890 − 0.454i)16-s + (0.968 − 0.250i)17-s + ⋯
L(s)  = 1  + (0.785 − 0.618i)2-s + (−0.900 + 0.433i)3-s + (0.233 − 0.972i)4-s + (0.995 − 0.0919i)5-s + (−0.439 + 0.898i)6-s + (0.932 + 0.359i)7-s + (−0.418 − 0.908i)8-s + (0.623 − 0.781i)9-s + (0.725 − 0.688i)10-s + (−0.919 − 0.391i)11-s + (0.211 + 0.977i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (−0.857 + 0.514i)15-s + (−0.890 − 0.454i)16-s + (0.968 − 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000102 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000102 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.000102 - 0.999i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.000102 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.344271613 - 1.344408827i\)
\(L(\frac12)\) \(\approx\) \(1.344271613 - 1.344408827i\)
\(L(1)\) \(\approx\) \(1.327547588 - 0.6286942179i\)
\(L(1)\) \(\approx\) \(1.327547588 - 0.6286942179i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.785 - 0.618i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (0.995 - 0.0919i)T \)
7 \( 1 + (0.932 + 0.359i)T \)
11 \( 1 + (-0.919 - 0.391i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (0.968 - 0.250i)T \)
19 \( 1 + (-0.910 - 0.413i)T \)
23 \( 1 + (0.709 - 0.705i)T \)
29 \( 1 + (0.188 + 0.982i)T \)
31 \( 1 + (0.386 - 0.922i)T \)
37 \( 1 + (-0.109 + 0.994i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.819 - 0.572i)T \)
47 \( 1 + (-0.0402 - 0.999i)T \)
53 \( 1 + (0.874 + 0.484i)T \)
59 \( 1 + (-0.200 - 0.979i)T \)
61 \( 1 + (0.973 + 0.228i)T \)
67 \( 1 + (0.740 - 0.671i)T \)
71 \( 1 + (0.343 - 0.939i)T \)
73 \( 1 + (-0.857 - 0.514i)T \)
79 \( 1 + (-0.0862 + 0.996i)T \)
83 \( 1 + (0.278 - 0.960i)T \)
89 \( 1 + (-0.289 + 0.957i)T \)
97 \( 1 + (-0.177 - 0.984i)T \)
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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

−23.341966088749214530806518384776, −23.18656215547576737855636269085, −21.887968538500995111600623250108, −21.24938520836056752609790486278, −20.88274808946179458586900567336, −19.2189596384579926420692873900, −18.176549615482082371017696077050, −17.34689312461169509441613241350, −17.10101457312491211802369905326, −16.10488141114894101140178113306, −14.913345631869823304227030216196, −14.19755803726878747310805314690, −13.3557822842478680792786231339, −12.60818930853923363363576012856, −11.77913166397020345413698459258, −10.78084113690986581998936093576, −9.94963347001844115481754411716, −8.38495837606355490132982263412, −7.43428022896092282787792866485, −6.77190138753650453162557937933, −5.631304627072403965300622531391, −5.15145211733733514842104396195, −4.23972022128602024786258212658, −2.50278571192149075594137174248, −1.61128446876650372294594655993, 0.88626009487183911161167541274, 2.157949686717672533327560198890, 3.144490435084047203495892273510, 4.7945595081098198297334891092, 5.10293184278951828999582131202, 5.86695108609724683910724295089, 6.90443861742504499746034327426, 8.469247469396496211078072840240, 9.7490376970009586561116539986, 10.39867861659386964070374946425, 11.03971545866314162616730633224, 12.054059575484466703624299302017, 12.78093375619809854125637478717, 13.61498844120300331919081644180, 14.822194118736506450227018762793, 15.16261146237577680133084101934, 16.49781788519308494589376979003, 17.261172486820940878829208337742, 18.27832105036800973911591298963, 18.72920434458835933688205128123, 20.34769491127520156025880607519, 20.92707713518757177573946682381, 21.61790839271111527246673039164, 22.03061368451962849191261377424, 23.09164783926377895315989680974

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