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Properties

Label 1-547-547.116-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.511 - 0.859i$
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  + (−0.857 − 0.514i)2-s + (−0.900 + 0.433i)3-s + (0.469 + 0.882i)4-s + (0.838 − 0.544i)5-s + (0.995 + 0.0919i)6-s + (−0.667 + 0.744i)7-s + (0.0517 − 0.998i)8-s + (0.623 − 0.781i)9-s + (−0.999 + 0.0345i)10-s + (0.987 + 0.160i)11-s + (−0.806 − 0.591i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (−0.519 + 0.854i)15-s + (−0.558 + 0.829i)16-s + (−0.999 + 0.0115i)17-s + ⋯
L(s)  = 1  + (−0.857 − 0.514i)2-s + (−0.900 + 0.433i)3-s + (0.469 + 0.882i)4-s + (0.838 − 0.544i)5-s + (0.995 + 0.0919i)6-s + (−0.667 + 0.744i)7-s + (0.0517 − 0.998i)8-s + (0.623 − 0.781i)9-s + (−0.999 + 0.0345i)10-s + (0.987 + 0.160i)11-s + (−0.806 − 0.591i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (−0.519 + 0.854i)15-s + (−0.558 + 0.829i)16-s + (−0.999 + 0.0115i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.511 - 0.859i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.511 - 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2137631688 - 0.3758824071i\)
\(L(\frac12)\) \(\approx\) \(0.2137631688 - 0.3758824071i\)
\(L(1)\) \(\approx\) \(0.5118264945 - 0.1392389084i\)
\(L(1)\) \(\approx\) \(0.5118264945 - 0.1392389084i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.857 - 0.514i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (0.838 - 0.544i)T \)
7 \( 1 + (-0.667 + 0.744i)T \)
11 \( 1 + (0.987 + 0.160i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (-0.999 + 0.0115i)T \)
19 \( 1 + (0.300 - 0.953i)T \)
23 \( 1 + (-0.177 - 0.984i)T \)
29 \( 1 + (-0.700 + 0.713i)T \)
31 \( 1 + (-0.868 - 0.495i)T \)
37 \( 1 + (-0.880 + 0.474i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.993 - 0.114i)T \)
47 \( 1 + (0.799 - 0.600i)T \)
53 \( 1 + (0.999 + 0.0230i)T \)
59 \( 1 + (-0.996 + 0.0804i)T \)
61 \( 1 + (-0.131 - 0.991i)T \)
67 \( 1 + (-0.109 + 0.994i)T \)
71 \( 1 + (0.973 + 0.228i)T \)
73 \( 1 + (-0.519 - 0.854i)T \)
79 \( 1 + (-0.596 - 0.802i)T \)
83 \( 1 + (-0.200 - 0.979i)T \)
89 \( 1 + (-0.792 - 0.609i)T \)
97 \( 1 + (-0.0632 + 0.997i)T \)
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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

−23.89043135747092576597232009523, −22.72407751886377309312103771890, −22.37004215230124565000808009795, −21.248422193515036258706783215309, −19.90865447000740272075379155188, −19.246515563563587412109665479651, −18.504695461673252385066675923164, −17.541474090166421933642197681700, −17.07896604496164865612831964380, −16.47826178449138236439796608232, −15.43971262888713785409797713340, −14.148970382929783047810134099062, −13.72666058256214313264349728759, −12.392692055536488552406390306744, −11.33728612073391046845688343614, −10.60764470332014906980847334600, −9.75741740885838971404914978982, −9.10596570402157424108392446065, −7.40916797039032721074774192292, −6.991453666324320696125141196410, −6.18266442330749488501765254482, −5.46012462129937419022165646362, −3.98234754139297348598880698068, −2.182036435394268746459776300519, −1.31534006267869039524224642291, 0.34943293096867233545165667597, 1.7520022248324673116936030720, 2.857776223115484951309658839503, 4.217411930408404037160879103964, 5.31542191371946321645018771390, 6.352989092914915329554152965764, 7.067684137711338703778041608168, 8.81219452185686766464217032984, 9.23832715154780289033017866174, 10.00581700233420520617133952692, 10.871238715932386737245253423565, 11.90425403145674424809935679592, 12.536182749478331629875626472527, 13.24111168222252894817290730427, 14.91723181219232099206594193629, 15.803151597897188686760857923564, 16.65779046039474899334735308110, 17.25092216899369083916852318665, 17.929109658783001229510205995565, 18.69831279989913587537704428607, 19.97274935105558228309139933982, 20.37620354610562533977133973138, 21.70731872254198616138226515462, 22.02917856326281128448814662676, 22.55682034327481595152043659992

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