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Properties

Label 1-539-539.222-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.602 + 0.797i$
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  + (−0.712 − 0.701i)2-s + (0.280 + 0.959i)3-s + (0.0149 + 0.999i)4-s + (0.646 + 0.762i)5-s + (0.473 − 0.880i)6-s + (0.691 − 0.722i)8-s + (−0.842 + 0.538i)9-s + (0.0747 − 0.997i)10-s + (−0.955 + 0.294i)12-s + (−0.963 + 0.266i)13-s + (−0.550 + 0.834i)15-s + (−0.999 + 0.0299i)16-s + (0.992 + 0.119i)17-s + (0.978 + 0.207i)18-s + (−0.978 + 0.207i)19-s + (−0.753 + 0.657i)20-s + ⋯
L(s)  = 1  + (−0.712 − 0.701i)2-s + (0.280 + 0.959i)3-s + (0.0149 + 0.999i)4-s + (0.646 + 0.762i)5-s + (0.473 − 0.880i)6-s + (0.691 − 0.722i)8-s + (−0.842 + 0.538i)9-s + (0.0747 − 0.997i)10-s + (−0.955 + 0.294i)12-s + (−0.963 + 0.266i)13-s + (−0.550 + 0.834i)15-s + (−0.999 + 0.0299i)16-s + (0.992 + 0.119i)17-s + (0.978 + 0.207i)18-s + (−0.978 + 0.207i)19-s + (−0.753 + 0.657i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.602 + 0.797i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.602 + 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3640992670 + 0.7313381626i\)
\(L(\frac12)\) \(\approx\) \(0.3640992670 + 0.7313381626i\)
\(L(1)\) \(\approx\) \(0.7345512645 + 0.2830617817i\)
\(L(1)\) \(\approx\) \(0.7345512645 + 0.2830617817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.712 - 0.701i)T \)
3 \( 1 + (0.280 + 0.959i)T \)
5 \( 1 + (0.646 + 0.762i)T \)
13 \( 1 + (-0.963 + 0.266i)T \)
17 \( 1 + (0.992 + 0.119i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (0.995 - 0.0896i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.575 + 0.817i)T \)
41 \( 1 + (-0.691 + 0.722i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.447 - 0.894i)T \)
53 \( 1 + (-0.599 - 0.800i)T \)
59 \( 1 + (-0.971 + 0.237i)T \)
61 \( 1 + (-0.599 + 0.800i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.753 + 0.657i)T \)
73 \( 1 + (-0.447 - 0.894i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (-0.963 - 0.266i)T \)
89 \( 1 + (-0.365 + 0.930i)T \)
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

−23.611697437550757166162227140535, −22.55766495242157471884224894607, −21.30576611922339715742607484638, −20.23699247070791742325937925687, −19.694228829556809851929138457750, −18.764874551553230232644750829170, −18.02189621449757256899648194589, −17.17000911095028100965112948832, −16.75910634827334574038481415935, −15.5737300451009345220770548154, −14.386603163976996240460535255871, −14.05572840663294733802434945520, −12.776543742779070231882217798353, −12.24998522620421040438192189441, −10.80259527711632150049512464477, −9.75655467984330263122391093227, −9.00731185659427814841488690225, −8.1203234251116666167161488731, −7.43297326108524596999356609516, −6.314397540471754820923185954065, −5.65841635810786723013670386922, −4.55642878888221869291535983935, −2.581070691920823924079678281267, −1.686444439219380671405278004554, −0.50289404477667227287264627748, 1.80266323935330350713521951220, 2.73511721011521667665947906911, 3.5686562017562075944947336941, 4.670985426689245081539486665794, 5.94161866726444728603796437985, 7.21610802333337743218435437636, 8.18670615560168293277964756299, 9.20935126115881742914895479149, 10.03141115069551521043450376199, 10.37247538009738764578260996279, 11.40414640583442462769151631308, 12.302780435316782564294725643244, 13.55127344397067499783034544169, 14.395215544489793325010410451058, 15.18997262891464773931836267723, 16.38374055402368967141565213235, 17.04060447522741823965737068862, 17.80577491492825523827307449782, 18.85356745073786203334661221327, 19.52199769691326994462312619188, 20.379479539921368735436130251593, 21.42645410311437480538123102317, 21.648172752906985446135666762982, 22.43713336235238049367237029849, 23.49231670075440885528249133088

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