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Properties

Label 1-539-539.173-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.979 + 0.202i$
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  + (0.447 − 0.894i)2-s + (0.337 − 0.941i)3-s + (−0.599 − 0.800i)4-s + (0.925 − 0.379i)5-s + (−0.691 − 0.722i)6-s + (−0.983 + 0.178i)8-s + (−0.772 − 0.635i)9-s + (0.0747 − 0.997i)10-s + (−0.955 + 0.294i)12-s + (−0.550 − 0.834i)13-s + (−0.0448 − 0.998i)15-s + (−0.280 + 0.959i)16-s + (0.420 − 0.907i)17-s + (−0.913 + 0.406i)18-s + (0.913 + 0.406i)19-s + (−0.858 − 0.512i)20-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)2-s + (0.337 − 0.941i)3-s + (−0.599 − 0.800i)4-s + (0.925 − 0.379i)5-s + (−0.691 − 0.722i)6-s + (−0.983 + 0.178i)8-s + (−0.772 − 0.635i)9-s + (0.0747 − 0.997i)10-s + (−0.955 + 0.294i)12-s + (−0.550 − 0.834i)13-s + (−0.0448 − 0.998i)15-s + (−0.280 + 0.959i)16-s + (0.420 − 0.907i)17-s + (−0.913 + 0.406i)18-s + (0.913 + 0.406i)19-s + (−0.858 − 0.512i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.979 + 0.202i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.979 + 0.202i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1867458551 - 1.824306919i\)
\(L(\frac12)\) \(\approx\) \(-0.1867458551 - 1.824306919i\)
\(L(1)\) \(\approx\) \(0.7601573301 - 1.234184977i\)
\(L(1)\) \(\approx\) \(0.7601573301 - 1.234184977i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.447 - 0.894i)T \)
3 \( 1 + (0.337 - 0.941i)T \)
5 \( 1 + (0.925 - 0.379i)T \)
13 \( 1 + (-0.550 - 0.834i)T \)
17 \( 1 + (0.420 - 0.907i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (-0.753 + 0.657i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.946 - 0.323i)T \)
41 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (-0.887 + 0.460i)T \)
53 \( 1 + (0.575 - 0.817i)T \)
59 \( 1 + (0.646 - 0.762i)T \)
61 \( 1 + (0.575 + 0.817i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (0.887 + 0.460i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.550 + 0.834i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
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

−24.06604851125645620625277043972, −22.75094943235796348661320849937, −22.25445474176553441937258817152, −21.440316833389237109125105872773, −21.02986768640995441289937740440, −19.804101267365697500363367167315, −18.65265956716231249143364168763, −17.67101674703546250655427198746, −16.8961223768955524507679442480, −16.25798319400372621227618571697, −15.28383569680549393801786177690, −14.40843786129323178590698956775, −14.08953256367390019219509439601, −13.08118410626264344448899311654, −11.94058500512154101441329434323, −10.72915747800299028214057087343, −9.703549268743688561199333419366, −9.151008892257282047783357509773, −8.08409211997982497736393164562, −7.0095587998567962828357620715, −5.96807489218098655206763724851, −5.21899789722437668436546936029, −4.226100956870121035644004046376, −3.2523907932706950008934540199, −2.15401495370662130622457370389, 0.79056285139727598475479125663, 1.83908367346535359375624792264, 2.67672995379899168082406085281, 3.68594111550338231598298115234, 5.397012452416293740308398780899, 5.60993851267342814409869648158, 7.03977331433473826567745256918, 8.112854129711585135648588271163, 9.33735863333255376328806298364, 9.77029314624399352724176661860, 11.02689201980260217793909293001, 12.084494437899917729590730946461, 12.66069516873610156923864215470, 13.48400802933019620172773836551, 14.112137408767163124920654868768, 14.8296098827349085839876004041, 16.24857262181160740313910833731, 17.61348719220862623783632717590, 17.9714994348576021595873297873, 18.85810312133150631161772690525, 19.8094240943625991529720972184, 20.48417796715801300268507660617, 21.04196113956368007184758496648, 22.31704729815562077865641636361, 22.68405523044426197930166699326

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