Properties

Label 1-483-483.419-r1-0-0
Degree $1$
Conductor $483$
Sign $0.744 + 0.667i$
Analytic cond. $51.9055$
Root an. cond. $51.9055$
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  + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (0.142 − 0.989i)8-s + (−0.654 − 0.755i)10-s + (0.841 − 0.540i)11-s + (0.654 + 0.755i)13-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + (0.415 + 0.909i)19-s + (0.142 + 0.989i)20-s − 22-s + (0.841 + 0.540i)25-s + (−0.142 − 0.989i)26-s + (−0.415 + 0.909i)29-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (0.142 − 0.989i)8-s + (−0.654 − 0.755i)10-s + (0.841 − 0.540i)11-s + (0.654 + 0.755i)13-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + (0.415 + 0.909i)19-s + (0.142 + 0.989i)20-s − 22-s + (0.841 + 0.540i)25-s + (−0.142 − 0.989i)26-s + (−0.415 + 0.909i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.744 + 0.667i$
Analytic conductor: \(51.9055\)
Root analytic conductor: \(51.9055\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (1:\ ),\ 0.744 + 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.586905093 + 0.6076755450i\)
\(L(\frac12)\) \(\approx\) \(1.586905093 + 0.6076755450i\)
\(L(1)\) \(\approx\) \(0.9692593555 + 0.009298906832i\)
\(L(1)\) \(\approx\) \(0.9692593555 + 0.009298906832i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
11 \( 1 + (0.841 - 0.540i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (0.142 - 0.989i)T \)
37 \( 1 + (0.959 - 0.281i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.654 + 0.755i)T \)
59 \( 1 + (-0.654 - 0.755i)T \)
61 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−23.650726641200881713165456304635, −22.67241232345142896193474479882, −21.83417899758766545298359475562, −20.41658608030309294977429788287, −20.285332207669077160833771967105, −19.02750874160234264355528278593, −18.002341974051678052843021525733, −17.60226282495233450344278695752, −16.76927433091595779694848230248, −15.824246451593149852650790989451, −15.026849262612817681078974565675, −13.96985782944434005019929992200, −13.3038328012918579121129993273, −11.9441204500455132084548768139, −10.94788294954818203301547887204, −9.95719271556238924674950303144, −9.26906178103957835302336631635, −8.54029894376930304480855884280, −7.2904210778069444672926165623, −6.48001299637755958316368598022, −5.568139310317513956265998656965, −4.62265461432436439995246678848, −2.804522237125685587125375535210, −1.6429885048497268144969764310, −0.63184193668280374711928753469, 1.21002380853807072519029160269, 1.92506411011560997228198025130, 3.219041038330310664833901213673, 4.17063968936610586811357352241, 5.9606039438354285233876120302, 6.54471368968597843410143878188, 7.778174299276379942989307798063, 8.91263030015175189076776182679, 9.4113996046310415728912451511, 10.49917343961675801634293605428, 11.164650008362490658415399920, 12.142162561753134809360832494300, 13.19299356267982761940269041658, 13.988138017709126624739163807348, 15.03125486629117654281414251150, 16.436380794495357330231143688685, 16.83218519141359787696463284890, 17.81063973642397599425304932885, 18.57245743267612387712603056019, 19.22709659874103993906956159072, 20.28599039357119410607553566569, 21.04560664191016624336366952799, 21.86485361245703354403546047959, 22.34946821454459656921153014516, 23.80434630549392682491389638211

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