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Properties

Label 1-4851-4851.1213-r0-0-0
Degree $1$
Conductor $4851$
Sign $-0.996 + 0.0891i$
Analytic cond. $22.5279$
Root an. cond. $22.5279$
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  + (−0.163 + 0.986i)2-s + (−0.946 − 0.323i)4-s + (−0.691 − 0.722i)5-s + (0.473 − 0.880i)8-s + (0.826 − 0.563i)10-s + (−0.163 + 0.986i)13-s + (0.791 + 0.611i)16-s + (0.0149 + 0.999i)17-s + (0.913 + 0.406i)19-s + (0.420 + 0.907i)20-s + (−0.222 + 0.974i)23-s + (−0.0448 + 0.998i)25-s + (−0.946 − 0.323i)26-s + (0.992 − 0.119i)29-s + (−0.978 + 0.207i)31-s + (−0.733 + 0.680i)32-s + ⋯
L(s)  = 1  + (−0.163 + 0.986i)2-s + (−0.946 − 0.323i)4-s + (−0.691 − 0.722i)5-s + (0.473 − 0.880i)8-s + (0.826 − 0.563i)10-s + (−0.163 + 0.986i)13-s + (0.791 + 0.611i)16-s + (0.0149 + 0.999i)17-s + (0.913 + 0.406i)19-s + (0.420 + 0.907i)20-s + (−0.222 + 0.974i)23-s + (−0.0448 + 0.998i)25-s + (−0.946 − 0.323i)26-s + (0.992 − 0.119i)29-s + (−0.978 + 0.207i)31-s + (−0.733 + 0.680i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4851\)    =    \(3^{2} \cdot 7^{2} \cdot 11\)
Sign: $-0.996 + 0.0891i$
Analytic conductor: \(22.5279\)
Root analytic conductor: \(22.5279\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4851} (1213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4851,\ (0:\ ),\ -0.996 + 0.0891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03615777768 + 0.8091501534i\)
\(L(\frac12)\) \(\approx\) \(0.03615777768 + 0.8091501534i\)
\(L(1)\) \(\approx\) \(0.6521743137 + 0.3913033540i\)
\(L(1)\) \(\approx\) \(0.6521743137 + 0.3913033540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.163 + 0.986i)T \)
5 \( 1 + (-0.691 - 0.722i)T \)
13 \( 1 + (-0.163 + 0.986i)T \)
17 \( 1 + (0.0149 + 0.999i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (0.992 - 0.119i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.992 - 0.119i)T \)
41 \( 1 + (0.525 + 0.850i)T \)
43 \( 1 + (0.0747 + 0.997i)T \)
47 \( 1 + (0.712 - 0.701i)T \)
53 \( 1 + (-0.873 + 0.486i)T \)
59 \( 1 + (0.525 - 0.850i)T \)
61 \( 1 + (0.0149 + 0.999i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.995 + 0.0896i)T \)
73 \( 1 + (0.251 - 0.967i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.163 - 0.986i)T \)
89 \( 1 + (-0.988 - 0.149i)T \)
97 \( 1 + (-0.978 + 0.207i)T \)
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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

−18.055774340814823321175919962825, −17.43937772249062822238665251109, −16.4120551404497523690603810884, −15.829186937370412515254996524839, −15.03262669179769397014391485185, −14.243329927317048969462621248635, −13.84028154931970456812699345142, −12.84426148819066754765421287100, −12.322290612084885038605350402102, −11.62515335310931754182514770524, −11.05757275521022715278650617023, −10.434622961981186863558235442419, −9.827883903600476987352850356881, −9.031660687102343722314619179028, −8.26362227056001811202626190655, −7.56133925498486169586493362540, −7.03923283106227421329696411370, −5.872073491562408159488898356208, −5.06318240534654940373597139742, −4.31879227333230516077506538441, −3.54753203246775402853472717127, −2.772938570061453293768058316293, −2.45074807982659346802311707718, −1.03987701924548096176170511421, −0.30389688591218978012603337056, 1.050681255073231040573323113221, 1.68985306744223949232029615663, 3.19825151059972865531657331615, 4.03213189515140116109725530381, 4.50227856350987275420785130018, 5.3368368745284545636892088820, 6.00776948275587041117692521014, 6.785658873161524033932698467729, 7.70373027821428456435178187180, 7.923876080012119151536280097038, 8.88516207898854162017589309835, 9.34572129911310565814077285714, 10.04518034324892648827587034054, 11.04692635309813447722525225816, 11.80204191205829636420821362714, 12.494935258049716535793694710468, 13.16413079554331814345618274536, 13.87252749596580057847035133366, 14.57905818158256173776786962261, 15.16614869294235851418656431192, 15.96324425302973117952159295891, 16.35515381863907958587446203256, 16.900318698808971404051032251027, 17.66793982179853853541226642592, 18.28015861940393078160577001583

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