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Properties

Label 1-4563-4563.1537-r0-0-0
Degree $1$
Conductor $4563$
Sign $-0.999 + 0.0265i$
Analytic cond. $21.1904$
Root an. cond. $21.1904$
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  + (−0.872 − 0.488i)2-s + (0.523 + 0.852i)4-s + (0.998 + 0.0536i)5-s + (−0.673 − 0.739i)7-s + (−0.0402 − 0.999i)8-s + (−0.845 − 0.534i)10-s + (0.653 − 0.757i)11-s + (0.226 + 0.974i)14-s + (−0.452 + 0.891i)16-s + (0.885 + 0.464i)17-s + (−0.5 + 0.866i)19-s + (0.476 + 0.879i)20-s + (−0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (0.994 + 0.107i)25-s + ⋯
L(s)  = 1  + (−0.872 − 0.488i)2-s + (0.523 + 0.852i)4-s + (0.998 + 0.0536i)5-s + (−0.673 − 0.739i)7-s + (−0.0402 − 0.999i)8-s + (−0.845 − 0.534i)10-s + (0.653 − 0.757i)11-s + (0.226 + 0.974i)14-s + (−0.452 + 0.891i)16-s + (0.885 + 0.464i)17-s + (−0.5 + 0.866i)19-s + (0.476 + 0.879i)20-s + (−0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (0.994 + 0.107i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $-0.999 + 0.0265i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ -0.999 + 0.0265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006205865318 - 0.4682165695i\)
\(L(\frac12)\) \(\approx\) \(0.006205865318 - 0.4682165695i\)
\(L(1)\) \(\approx\) \(0.6707686098 - 0.2260074242i\)
\(L(1)\) \(\approx\) \(0.6707686098 - 0.2260074242i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.872 - 0.488i)T \)
5 \( 1 + (0.998 + 0.0536i)T \)
7 \( 1 + (-0.673 - 0.739i)T \)
11 \( 1 + (0.653 - 0.757i)T \)
17 \( 1 + (0.885 + 0.464i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.872 - 0.488i)T \)
31 \( 1 + (-0.589 - 0.807i)T \)
37 \( 1 + (0.278 + 0.960i)T \)
41 \( 1 + (-0.711 - 0.702i)T \)
43 \( 1 + (-0.589 + 0.807i)T \)
47 \( 1 + (-0.999 - 0.0268i)T \)
53 \( 1 + (0.885 + 0.464i)T \)
59 \( 1 + (-0.452 - 0.891i)T \)
61 \( 1 + (-0.991 + 0.133i)T \)
67 \( 1 + (-0.783 + 0.621i)T \)
71 \( 1 + (0.568 - 0.822i)T \)
73 \( 1 + (0.987 - 0.160i)T \)
79 \( 1 + (0.476 + 0.879i)T \)
83 \( 1 + (-0.252 - 0.967i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.545 - 0.837i)T \)
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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

−18.287524739854565501440035045159, −18.02805626480606941528675445583, −17.23246769918978376662006236237, −16.51040455977084621092868528389, −16.20764166630589753187639587913, −15.09762885195855488362054359448, −14.79646969281638122566525353168, −14.01384385384596087217062637254, −13.22647574089897543789070711742, −12.37144194001919798674518702252, −11.836361227599569688569151448039, −10.79435202950627926164920265257, −10.16762720434826072789099047538, −9.440329390881736889467758698345, −9.20055355417551882032440222604, −8.43987696335603053859548806965, −7.44696139620654344808018935480, −6.69591188119798548489203021920, −6.294287240984745331722137329165, −5.411832274966439999417852238927, −4.96730506178256210362782656095, −3.6186423157090450564376966431, −2.56418832700833150455227689149, −2.001729800520597995747955588612, −1.20105892405348436632088031955, 0.16334120788141101887675632032, 1.3744741643545338606031041903, 1.76816754750029113798615308122, 2.90954834207377474045820032807, 3.57907456014166999196331673810, 4.159345215182315785039759374442, 5.607265560947225133526562664875, 6.224188631626623841976440470991, 6.738174933056325600665610237021, 7.771190081656674064222721773546, 8.255983886703101755698460538886, 9.29910684999325118486537917639, 9.67544003687200611930932700144, 10.29773225453440769204061969799, 10.849043936282812564579858404038, 11.71359433306997876154367316935, 12.409958589521008220551675925065, 13.19179198983179470037142753997, 13.64981147271711463110113438540, 14.45664497494120241113618844742, 15.26705794790657009152945952362, 16.43202226629944755631947955344, 16.73143058833554823063872512202, 17.043327212439532839111181096984, 17.95040827514113542968632649931

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