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Properties

Label 1-539-539.194-r0-0-0
Degree $1$
Conductor $539$
Sign $0.999 + 0.0330i$
Analytic cond. $2.50310$
Root an. cond. $2.50310$
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  + (0.280 − 0.959i)2-s + (−0.193 − 0.981i)3-s + (−0.842 − 0.538i)4-s + (−0.0149 + 0.999i)5-s + (−0.995 − 0.0896i)6-s + (−0.753 + 0.657i)8-s + (−0.925 + 0.379i)9-s + (0.955 + 0.294i)10-s + (−0.365 + 0.930i)12-s + (−0.691 − 0.722i)13-s + (0.983 − 0.178i)15-s + (0.420 + 0.907i)16-s + (−0.163 + 0.986i)17-s + (0.104 + 0.994i)18-s + (−0.104 + 0.994i)19-s + (0.550 − 0.834i)20-s + ⋯
L(s)  = 1  + (0.280 − 0.959i)2-s + (−0.193 − 0.981i)3-s + (−0.842 − 0.538i)4-s + (−0.0149 + 0.999i)5-s + (−0.995 − 0.0896i)6-s + (−0.753 + 0.657i)8-s + (−0.925 + 0.379i)9-s + (0.955 + 0.294i)10-s + (−0.365 + 0.930i)12-s + (−0.691 − 0.722i)13-s + (0.983 − 0.178i)15-s + (0.420 + 0.907i)16-s + (−0.163 + 0.986i)17-s + (0.104 + 0.994i)18-s + (−0.104 + 0.994i)19-s + (0.550 − 0.834i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.999 + 0.0330i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.999 + 0.0330i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7800909922 + 0.01289181611i\)
\(L(\frac12)\) \(\approx\) \(0.7800909922 + 0.01289181611i\)
\(L(1)\) \(\approx\) \(0.7621139374 - 0.3798035142i\)
\(L(1)\) \(\approx\) \(0.7621139374 - 0.3798035142i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.280 - 0.959i)T \)
3 \( 1 + (-0.193 - 0.981i)T \)
5 \( 1 + (-0.0149 + 0.999i)T \)
13 \( 1 + (-0.691 - 0.722i)T \)
17 \( 1 + (-0.163 + 0.986i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.988 - 0.149i)T \)
29 \( 1 + (0.963 + 0.266i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.251 + 0.967i)T \)
41 \( 1 + (0.753 - 0.657i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (0.337 + 0.941i)T \)
53 \( 1 + (-0.772 + 0.635i)T \)
59 \( 1 + (0.946 - 0.323i)T \)
61 \( 1 + (-0.772 - 0.635i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.550 - 0.834i)T \)
73 \( 1 + (-0.337 + 0.941i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.691 + 0.722i)T \)
89 \( 1 + (-0.0747 - 0.997i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−23.48983939213292296642136886193, −22.6401110561329916739930432064, −21.74383796318476720813679454167, −21.24097071845320989068371466998, −20.24816036814214428563629900951, −19.34041856893041424285762544416, −17.82304783719387694701417070029, −17.338181287345095371573048583707, −16.33922165718506467917910389906, −16.00426420418204960076306912451, −15.13679658758086842918622533735, −14.123265181242841234348336053055, −13.451573420723588903375436163006, −12.18702573855366062314008885383, −11.62231693388005100506164468900, −10.04685435137303409481871451898, −9.28475451291086594567440804103, −8.6855904309146612275389676270, −7.59416075616082974409383557324, −6.42091219142921505848405746088, −5.4047853632949678325119419669, −4.63445535121661540878210896213, −4.11401855827740433340767749213, −2.64906051698367106428857345981, −0.4033376731779168933937278773, 1.32473444644063599031135774763, 2.41452527303418032410852795621, 3.13094182814599483331976394109, 4.39907392875786800171966324935, 5.79677528158041887346920506715, 6.35880928609123171083886103499, 7.70043100966275653282810994544, 8.41257061580548943485297407140, 9.958613653897082369268089857063, 10.52063183714207781047411785164, 11.47277106062766909812989891148, 12.300796900728486745425924806551, 12.8976562500427327331001202972, 14.08688634977012041154759387278, 14.41661673992123058390172613872, 15.53209939376745524342499059637, 17.184112990362457650335323372197, 17.79816948739261265642397984298, 18.53076961455281927791201216628, 19.33145495644401090339576581686, 19.78576344238500234781263801854, 20.90568696822764338330643195508, 22.00820272774099565760344761558, 22.49107869585317025336187015556, 23.27214213666345009045800721681

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