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Properties

Label 1-547-547.136-r0-0-0
Degree $1$
Conductor $547$
Sign $0.998 - 0.0515i$
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  + (−0.200 + 0.979i)2-s + 3-s + (−0.919 − 0.391i)4-s + (−0.996 + 0.0804i)5-s + (−0.200 + 0.979i)6-s + (0.948 + 0.316i)7-s + (0.568 − 0.822i)8-s + 9-s + (0.120 − 0.992i)10-s + (−0.0402 − 0.999i)11-s + (−0.919 − 0.391i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.996 + 0.0804i)15-s + (0.692 + 0.721i)16-s + (−0.845 − 0.534i)17-s + ⋯
L(s)  = 1  + (−0.200 + 0.979i)2-s + 3-s + (−0.919 − 0.391i)4-s + (−0.996 + 0.0804i)5-s + (−0.200 + 0.979i)6-s + (0.948 + 0.316i)7-s + (0.568 − 0.822i)8-s + 9-s + (0.120 − 0.992i)10-s + (−0.0402 − 0.999i)11-s + (−0.919 − 0.391i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.996 + 0.0804i)15-s + (0.692 + 0.721i)16-s + (−0.845 − 0.534i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.998 - 0.0515i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.998 - 0.0515i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.325560450 - 0.03417795469i\)
\(L(\frac12)\) \(\approx\) \(1.325560450 - 0.03417795469i\)
\(L(1)\) \(\approx\) \(1.073566977 + 0.2471216180i\)
\(L(1)\) \(\approx\) \(1.073566977 + 0.2471216180i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.200 + 0.979i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.996 + 0.0804i)T \)
7 \( 1 + (0.948 + 0.316i)T \)
11 \( 1 + (-0.0402 - 0.999i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.845 - 0.534i)T \)
19 \( 1 + (0.692 - 0.721i)T \)
23 \( 1 + (-0.632 - 0.774i)T \)
29 \( 1 + (-0.354 - 0.935i)T \)
31 \( 1 + (-0.970 + 0.239i)T \)
37 \( 1 + (-0.632 + 0.774i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.799 - 0.600i)T \)
47 \( 1 + (0.987 + 0.160i)T \)
53 \( 1 + (0.428 - 0.903i)T \)
59 \( 1 + (0.692 + 0.721i)T \)
61 \( 1 + (-0.200 + 0.979i)T \)
67 \( 1 + (0.799 - 0.600i)T \)
71 \( 1 + (0.278 + 0.960i)T \)
73 \( 1 + (-0.996 - 0.0804i)T \)
79 \( 1 + (0.120 + 0.992i)T \)
83 \( 1 + (0.428 - 0.903i)T \)
89 \( 1 + (-0.748 + 0.663i)T \)
97 \( 1 + (-0.0402 + 0.999i)T \)
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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

−23.52966633418340837410415086614, −22.28924074348833902645563317387, −21.50690105213577154017546410694, −20.52411679819002321004452170061, −20.114929895557275313813369227505, −19.493509344500611993390839607791, −18.54363541925784408010980463789, −17.85640192826262911729569955469, −16.75295363418057997886643882942, −15.59593902447337762012765530625, −14.599304104452321581825808447, −14.13337077664888750181152173502, −12.94098618804530149295577175140, −12.19600642839743589486495030766, −11.36332150069666702314097699207, −10.43867527565888669436128148967, −9.41721930282642588555617756702, −8.67669224171458183264325458094, −7.67382162033616723887832843610, −7.28791087377604462430559564898, −4.971728513195216738172584429728, −4.185597724638994210208555320383, −3.59686184715752784032030325405, −2.1611063959214725106752719276, −1.50399352087509923070458835950, 0.70715651474270089553119086285, 2.454454847373007972816115546390, 3.670465192217057613153706361243, 4.596823766167900756552461870979, 5.527603678288491863268963196119, 7.03963899308327315380240105137, 7.626674281786218071538104896428, 8.50838604955805199419151263032, 8.86570159612041718263472525996, 10.20620876950994642211931108322, 11.226237883896497386841664032711, 12.39405876272666283020232931817, 13.53086494998453076717600301913, 14.17892042198529755865095292805, 15.10456994328484933097092328145, 15.53391136372718283579977433564, 16.31638563202154473090615604107, 17.57095850661496765095336425178, 18.390412515921768956647473164047, 19.03712017583778725250473629560, 19.94056829550119196679486763386, 20.64603759100363198688920309686, 21.98298181191330428800537412649, 22.48801997210962684137736854893, 23.977625352347675280831965495

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