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Properties

Label 1-539-539.158-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.186 - 0.982i$
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  + (0.971 − 0.237i)2-s + (−0.946 − 0.323i)3-s + (0.887 − 0.460i)4-s + (−0.873 + 0.486i)5-s + (−0.995 − 0.0896i)6-s + (0.753 − 0.657i)8-s + (0.791 + 0.611i)9-s + (−0.733 + 0.680i)10-s + (−0.988 + 0.149i)12-s + (−0.691 − 0.722i)13-s + (0.983 − 0.178i)15-s + (0.575 − 0.817i)16-s + (−0.772 − 0.635i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + (−0.550 + 0.834i)20-s + ⋯
L(s)  = 1  + (0.971 − 0.237i)2-s + (−0.946 − 0.323i)3-s + (0.887 − 0.460i)4-s + (−0.873 + 0.486i)5-s + (−0.995 − 0.0896i)6-s + (0.753 − 0.657i)8-s + (0.791 + 0.611i)9-s + (−0.733 + 0.680i)10-s + (−0.988 + 0.149i)12-s + (−0.691 − 0.722i)13-s + (0.983 − 0.178i)15-s + (0.575 − 0.817i)16-s + (−0.772 − 0.635i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + (−0.550 + 0.834i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.186 - 0.982i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.186 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8659323608 - 1.045994435i\)
\(L(\frac12)\) \(\approx\) \(0.8659323608 - 1.045994435i\)
\(L(1)\) \(\approx\) \(1.108615305 - 0.4366759583i\)
\(L(1)\) \(\approx\) \(1.108615305 - 0.4366759583i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.971 - 0.237i)T \)
3 \( 1 + (-0.946 - 0.323i)T \)
5 \( 1 + (-0.873 + 0.486i)T \)
13 \( 1 + (-0.691 - 0.722i)T \)
17 \( 1 + (-0.772 - 0.635i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.365 + 0.930i)T \)
29 \( 1 + (-0.963 - 0.266i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.712 - 0.701i)T \)
41 \( 1 + (0.753 - 0.657i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.646 + 0.762i)T \)
53 \( 1 + (-0.163 - 0.986i)T \)
59 \( 1 + (0.193 - 0.981i)T \)
61 \( 1 + (-0.163 + 0.986i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.550 - 0.834i)T \)
73 \( 1 + (-0.646 - 0.762i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.691 + 0.722i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
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

−23.49738262145584867905429591596, −22.91500316655729223208217771324, −22.11427216653565366059628418013, −21.423821963869878664745469101117, −20.486919842447473460517647563858, −19.74161917131090539366508233274, −18.62423690638879270773233872805, −17.381924736636174978419076013489, −16.57800138720320402899280690261, −16.12560949454839915525259433300, −15.19425383943151885969622267974, −14.51338805759260736836656723986, −13.12848120966024573419570928114, −12.48280586826750301768918536122, −11.65987688440041484313967883828, −11.16188808956385529990664176112, −9.995942902702051869607897453984, −8.68028916749493550745744684769, −7.49855770400049126868315377838, −6.72476631811929480084633916649, −5.730307910962694295216935807800, −4.636446916933984005888959646718, −4.311111906510227113666637251454, −3.07130014190616397548447788076, −1.42141397323507834849455163176, 0.60947514933641504288224210310, 2.21334934167846930535612522301, 3.29948360292280355654923323060, 4.430350100528560591081642360743, 5.22809672795817101543233071029, 6.179429912573933623588895809597, 7.301756635725074043481648981808, 7.58412644903287268408308488760, 9.581483050629631950624459640831, 10.63875347470624612738069567995, 11.4121518732591142264649020483, 11.84154209990930327586632777590, 12.84896541858034839204861442357, 13.55909352186538389031868040182, 14.73966552541004470290380905446, 15.57262104867418853575468197319, 16.09675492778755434841550624920, 17.26951299399864392420393651761, 18.19409397751513677227207877451, 19.18045316239009202908560022685, 19.81079604992319561035417688511, 20.77517128823561666367167230552, 21.945579416247552113064835089854, 22.56018484763665117882291770250, 22.88084628006997649813897923459

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