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Properties

Label 1-547-547.508-r0-0-0
Degree $1$
Conductor $547$
Sign $0.0152 + 0.999i$
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  + (0.365 − 0.930i)2-s + (−0.222 + 0.974i)3-s + (−0.733 − 0.680i)4-s + (0.826 − 0.563i)5-s + (0.826 + 0.563i)6-s + (−0.733 + 0.680i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.222 − 0.974i)10-s + (−0.5 − 0.866i)11-s + (0.826 − 0.563i)12-s + (−0.988 − 0.149i)13-s + (0.365 + 0.930i)14-s + (0.365 + 0.930i)15-s + (0.0747 + 0.997i)16-s + (0.0747 + 0.997i)17-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)2-s + (−0.222 + 0.974i)3-s + (−0.733 − 0.680i)4-s + (0.826 − 0.563i)5-s + (0.826 + 0.563i)6-s + (−0.733 + 0.680i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.222 − 0.974i)10-s + (−0.5 − 0.866i)11-s + (0.826 − 0.563i)12-s + (−0.988 − 0.149i)13-s + (0.365 + 0.930i)14-s + (0.365 + 0.930i)15-s + (0.0747 + 0.997i)16-s + (0.0747 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0152 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0152 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.0152 + 0.999i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.0152 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4295174835 + 0.4230268368i\)
\(L(\frac12)\) \(\approx\) \(0.4295174835 + 0.4230268368i\)
\(L(1)\) \(\approx\) \(0.8440225283 - 0.08760522593i\)
\(L(1)\) \(\approx\) \(0.8440225283 - 0.08760522593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.365 - 0.930i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.826 - 0.563i)T \)
7 \( 1 + (-0.733 + 0.680i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
17 \( 1 + (0.0747 + 0.997i)T \)
19 \( 1 + (0.365 + 0.930i)T \)
23 \( 1 + (0.365 + 0.930i)T \)
29 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (-0.222 + 0.974i)T \)
37 \( 1 + (0.0747 + 0.997i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.733 + 0.680i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.988 - 0.149i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.0747 + 0.997i)T \)
67 \( 1 + (0.0747 + 0.997i)T \)
71 \( 1 + (0.0747 + 0.997i)T \)
73 \( 1 + (0.365 - 0.930i)T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (0.365 + 0.930i)T \)
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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

−23.08654143085943445127128817288, −22.6220117614884600244816445828, −21.99342780308950906044894456856, −20.70204441170152486781852683073, −19.741254340348837543089892131224, −18.49917945010594319550035838991, −18.165048505921634841822767191, −17.03581556031646314220446273154, −16.82195083913841981685802924503, −15.43202240921031892249414518435, −14.520654659142619532002559664673, −13.738643211577752218045078254787, −13.114673752530720507710411151659, −12.45465289836241240294655154645, −11.240163755576941299484688430168, −9.90411031746087520191912286666, −9.27193397164941167994366953969, −7.735614712477143573331050063284, −7.06969902298493779995963791490, −6.65607380181983169591421196647, −5.52754946057870019071591548452, −4.69805042013882128303768321271, −3.07780996220737468405075414227, −2.248666156556912549642763911266, −0.27148658799457209562889572135, 1.57718920913651033916286927280, 2.89092550677656532898317219775, 3.54744319536207831433827267084, 4.93852664678357502198218119318, 5.51951605513133711837060113036, 6.18660437735147530600755082281, 8.34733521048713494344171584391, 9.16938401991379606299872558945, 9.87537280382295236355596765654, 10.45193610591450103807719889823, 11.55426744280455798481323785529, 12.46701888367265619272625413496, 13.126025320205324036134617333398, 14.1867174893000960417553688281, 14.975650684529798168389619271359, 15.97566717703277455608464920169, 16.85603319114608696547355326864, 17.68294811110735964341013548126, 18.76369300466021381314854551329, 19.577826177955597118282375582822, 20.49645124777406313040219203982, 21.240766568047905622162286964625, 21.90502691097355415519850111722, 22.2096065268740668455278187832, 23.36626141931602830057485449440

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