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Properties

Label 1-547-547.143-r0-0-0
Degree $1$
Conductor $547$
Sign $0.722 + 0.690i$
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  + (−0.577 − 0.816i)2-s + (−0.222 − 0.974i)3-s + (−0.332 + 0.942i)4-s + (−0.244 − 0.969i)5-s + (−0.667 + 0.744i)6-s + (0.548 + 0.835i)7-s + (0.962 − 0.272i)8-s + (−0.900 + 0.433i)9-s + (−0.650 + 0.759i)10-s + (−0.632 − 0.774i)11-s + (0.993 + 0.114i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.890 + 0.454i)15-s + (−0.778 − 0.627i)16-s + (−0.958 + 0.283i)17-s + ⋯
L(s)  = 1  + (−0.577 − 0.816i)2-s + (−0.222 − 0.974i)3-s + (−0.332 + 0.942i)4-s + (−0.244 − 0.969i)5-s + (−0.667 + 0.744i)6-s + (0.548 + 0.835i)7-s + (0.962 − 0.272i)8-s + (−0.900 + 0.433i)9-s + (−0.650 + 0.759i)10-s + (−0.632 − 0.774i)11-s + (0.993 + 0.114i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.890 + 0.454i)15-s + (−0.778 − 0.627i)16-s + (−0.958 + 0.283i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.722 + 0.690i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.722 + 0.690i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1423377111 + 0.05707618456i\)
\(L(\frac12)\) \(\approx\) \(0.1423377111 + 0.05707618456i\)
\(L(1)\) \(\approx\) \(0.4250172873 - 0.3097955466i\)
\(L(1)\) \(\approx\) \(0.4250172873 - 0.3097955466i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.577 - 0.816i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.244 - 0.969i)T \)
7 \( 1 + (0.548 + 0.835i)T \)
11 \( 1 + (-0.632 - 0.774i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (-0.958 + 0.283i)T \)
19 \( 1 + (0.973 - 0.228i)T \)
23 \( 1 + (0.968 + 0.250i)T \)
29 \( 1 + (-0.539 + 0.842i)T \)
31 \( 1 + (-0.928 - 0.370i)T \)
37 \( 1 + (-0.980 - 0.194i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.965 - 0.261i)T \)
47 \( 1 + (-0.919 + 0.391i)T \)
53 \( 1 + (0.838 + 0.544i)T \)
59 \( 1 + (0.428 + 0.903i)T \)
61 \( 1 + (0.166 + 0.986i)T \)
67 \( 1 + (-0.397 - 0.917i)T \)
71 \( 1 + (0.863 - 0.504i)T \)
73 \( 1 + (-0.890 - 0.454i)T \)
79 \( 1 + (0.256 + 0.966i)T \)
83 \( 1 + (0.948 - 0.316i)T \)
89 \( 1 + (0.770 + 0.636i)T \)
97 \( 1 + (-0.999 - 0.0115i)T \)
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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

−23.102368859451023099142421936820, −22.722671237291428370338099536618, −21.86147526263377801290657861063, −20.527629772601619682421828226000, −20.03571815260053099857060968811, −18.935003592772382999355778838660, −17.827942176505273761336843092408, −17.4908738727690802082771197789, −16.5083559473360863174337152529, −15.59441562594384152767452354845, −14.942664742386494203334140464479, −14.403847904592748060041624966053, −13.39372656787074223574499096809, −11.67313545422538436446988245875, −10.86532906947928934614945871048, −10.18028278060887785291909900618, −9.57427816633686495831839868041, −8.31012355558129819287474345711, −7.30443098517295134017279933163, −6.81546074212828654823279850936, −5.301858683155993059014507218346, −4.7593658413270220325551778081, −3.58871386054228491607937651251, −2.14873111490269716010869897196, −0.10344477817512687611534148752, 1.28702737640486427131152765848, 2.174830747045116800548191775689, 3.21261778868294518119344358713, 4.869335283356635563969513471586, 5.44692636133640741094430964475, 7.0857382496165100380667132100, 7.921750217747302030534020679432, 8.70233203801317516399788567462, 9.29119380860608733614632083363, 10.83598776214844925027036016043, 11.57706356571372490531561368535, 12.1830231623350725536106217751, 13.02694051877225721921353873097, 13.590003131167322914132170825650, 15.00066591293026229184541541476, 16.27270044458366086031409599884, 16.94926704938286632437637114463, 17.869569597106995596525771914496, 18.4173180608917263387509981110, 19.36099768324008598357494027867, 19.91382955387821010854074965758, 20.83681947517054811568680906600, 21.71627971378166853035000027626, 22.42079708975193200757599749779, 23.71172355251914230302760618911

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