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Properties

Label 1-539-539.206-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.472 - 0.881i$
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  + (−0.575 + 0.817i)2-s + (−0.251 − 0.967i)3-s + (−0.337 − 0.941i)4-s + (0.447 + 0.894i)5-s + (0.936 + 0.351i)6-s + (0.963 + 0.266i)8-s + (−0.873 + 0.486i)9-s + (−0.988 − 0.149i)10-s + (−0.826 + 0.563i)12-s + (−0.995 − 0.0896i)13-s + (0.753 − 0.657i)15-s + (−0.772 + 0.635i)16-s + (−0.925 + 0.379i)17-s + (0.104 − 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.691 − 0.722i)20-s + ⋯
L(s)  = 1  + (−0.575 + 0.817i)2-s + (−0.251 − 0.967i)3-s + (−0.337 − 0.941i)4-s + (0.447 + 0.894i)5-s + (0.936 + 0.351i)6-s + (0.963 + 0.266i)8-s + (−0.873 + 0.486i)9-s + (−0.988 − 0.149i)10-s + (−0.826 + 0.563i)12-s + (−0.995 − 0.0896i)13-s + (0.753 − 0.657i)15-s + (−0.772 + 0.635i)16-s + (−0.925 + 0.379i)17-s + (0.104 − 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.691 − 0.722i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.472 - 0.881i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.472 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1519195113 - 0.2539400946i\)
\(L(\frac12)\) \(\approx\) \(0.1519195113 - 0.2539400946i\)
\(L(1)\) \(\approx\) \(0.5733034430 + 0.02382741871i\)
\(L(1)\) \(\approx\) \(0.5733034430 + 0.02382741871i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.575 + 0.817i)T \)
3 \( 1 + (-0.251 - 0.967i)T \)
5 \( 1 + (0.447 + 0.894i)T \)
13 \( 1 + (-0.995 - 0.0896i)T \)
17 \( 1 + (-0.925 + 0.379i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (-0.473 - 0.880i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.999 - 0.0299i)T \)
41 \( 1 + (-0.963 - 0.266i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.946 - 0.323i)T \)
53 \( 1 + (0.791 - 0.611i)T \)
59 \( 1 + (-0.712 - 0.701i)T \)
61 \( 1 + (0.791 + 0.611i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.691 - 0.722i)T \)
73 \( 1 + (-0.946 - 0.323i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (-0.995 + 0.0896i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
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

−23.53330104876618785263268511941, −22.446783344258822666037804309846, −21.80770905363640067944551914956, −21.15373865046019161347425801650, −20.32178126131293414432049461958, −19.87609662533983004875704805648, −18.70812804900871186760542004945, −17.494563519904030277234251543322, −17.172061442375670935117728551271, −16.32428571179563808500204121149, −15.49497042681586970185015497345, −14.19312499786878898191853512378, −13.28934882886226347984842323949, −12.19320681783110475879585963939, −11.68193780772745516600192357310, −10.46363092215019975447845439134, −9.88904484078732101536260292021, −9.06073422413475995240080938695, −8.44019348927183904203349021828, −7.10427646549714563020116633789, −5.55817160214411079127978656653, −4.74209449786911568336703640546, −3.87553992789204261014853737043, −2.67165828241774045214263583077, −1.46383411602925702206970528144, 0.1902635918268910306755811774, 1.85736202268100662419971826295, 2.668004146548519989668015465675, 4.624761067904837472797199018211, 5.69033119499022657767238692223, 6.64275900784132351939771813439, 7.014429626991844258295169631993, 8.0416130543822052888081059747, 8.985880708530800900008103795419, 10.11766218678833585069201457615, 10.87566217607631908449341840202, 11.862303208399506030724929412824, 13.210551466947965812542126763363, 13.7626862001112063183215590397, 14.76684848503864273417010906920, 15.35129336742116690752401322641, 16.71992506639582116069600642733, 17.48043880727132464100214346039, 17.857116008269284329408383933824, 18.908494925580409614447485502591, 19.305964455775432497281229180710, 20.29174123751481485608270498090, 21.90897234713058760825502626695, 22.51430784912120918719607114315, 23.25562858251905387164989266194

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