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Properties

Label 1-539-539.9-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.931 + 0.362i$
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  + (0.525 − 0.850i)2-s + (0.575 − 0.817i)3-s + (−0.447 − 0.894i)4-s + (0.193 − 0.981i)5-s + (−0.393 − 0.919i)6-s + (−0.995 − 0.0896i)8-s + (−0.337 − 0.941i)9-s + (−0.733 − 0.680i)10-s + (−0.988 − 0.149i)12-s + (0.473 + 0.880i)13-s + (−0.691 − 0.722i)15-s + (−0.599 + 0.800i)16-s + (−0.842 − 0.538i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + (−0.963 + 0.266i)20-s + ⋯
L(s)  = 1  + (0.525 − 0.850i)2-s + (0.575 − 0.817i)3-s + (−0.447 − 0.894i)4-s + (0.193 − 0.981i)5-s + (−0.393 − 0.919i)6-s + (−0.995 − 0.0896i)8-s + (−0.337 − 0.941i)9-s + (−0.733 − 0.680i)10-s + (−0.988 − 0.149i)12-s + (0.473 + 0.880i)13-s + (−0.691 − 0.722i)15-s + (−0.599 + 0.800i)16-s + (−0.842 − 0.538i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + (−0.963 + 0.266i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.931 + 0.362i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.931 + 0.362i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3371085939 - 1.796492378i\)
\(L(\frac12)\) \(\approx\) \(-0.3371085939 - 1.796492378i\)
\(L(1)\) \(\approx\) \(0.7341189012 - 1.270397129i\)
\(L(1)\) \(\approx\) \(0.7341189012 - 1.270397129i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.525 - 0.850i)T \)
3 \( 1 + (0.575 - 0.817i)T \)
5 \( 1 + (0.193 - 0.981i)T \)
13 \( 1 + (0.473 + 0.880i)T \)
17 \( 1 + (-0.842 - 0.538i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.365 - 0.930i)T \)
29 \( 1 + (0.936 + 0.351i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (-0.163 - 0.986i)T \)
41 \( 1 + (-0.995 - 0.0896i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.971 + 0.237i)T \)
53 \( 1 + (0.887 + 0.460i)T \)
59 \( 1 + (0.420 - 0.907i)T \)
61 \( 1 + (0.887 - 0.460i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.963 - 0.266i)T \)
73 \( 1 + (0.971 - 0.237i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.473 - 0.880i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−23.768738338344681940552070460, −22.96239358487639620071123366856, −22.21360942374660302576943146022, −21.594813615570565921616668818673, −20.890745436128747172678325105519, −19.75889556400076849268413960885, −18.83539749867474522839784312571, −17.67607494645639630827972275714, −17.14451446150745164579738297336, −15.87754071227108481264861197486, −15.2050171939530248891670959943, −14.90685836235949587115017193052, −13.5903470986995693341716542805, −13.435675006022098580907616402, −11.88007620125639710956614489152, −10.74678352698909474907408506069, −10.05703703670612598378984353469, −8.78049010121711588958942309676, −8.1899306458053554289505524270, −7.04659840091032083910130887455, −6.166316450468578183417426440007, −5.15840319015047580178925112626, −4.07899363834452733797131511550, −3.26699340440556482166932952172, −2.37860000244260360495020696700, 0.74188142172548619480230308093, 1.82959125297012925722349112950, 2.60691239381530094082415732892, 3.9884878174150383303472873424, 4.73611478591193192439944321574, 6.04625972159386216064147916498, 6.8156008601066017195750474682, 8.47803652447425237096808907259, 8.843191475104876903071692706444, 9.839951951464743208173460715753, 11.09436830486565553115678762376, 12.03474553589852903687959684201, 12.67000679685070166276558149701, 13.46328054090502168965558001826, 14.017319211680732564649466923232, 15.00078501003331681216441482020, 16.083304019346131964429111008949, 17.28993751829914671324956719169, 18.15675177889650233274967643051, 19.037341967771972144597695923911, 19.65995545055476129014203054556, 20.57745212421590253295579919864, 20.97481104219807195849947734222, 21.939331194304931142809547664256, 23.20685638367061136051091908198

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