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Properties

Label 1-4563-4563.250-r0-0-0
Degree $1$
Conductor $4563$
Sign $-0.963 + 0.267i$
Analytic cond. $21.1904$
Root an. cond. $21.1904$
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  + (0.909 + 0.416i)2-s + (0.653 + 0.757i)4-s + (0.930 + 0.367i)5-s + (−0.897 + 0.440i)7-s + (0.278 + 0.960i)8-s + (0.692 + 0.721i)10-s + (0.964 + 0.265i)11-s + (−0.999 + 0.0268i)14-s + (−0.147 + 0.989i)16-s + (−0.970 − 0.239i)17-s + (−0.5 + 0.866i)19-s + (0.329 + 0.944i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + (0.730 + 0.682i)25-s + ⋯
L(s)  = 1  + (0.909 + 0.416i)2-s + (0.653 + 0.757i)4-s + (0.930 + 0.367i)5-s + (−0.897 + 0.440i)7-s + (0.278 + 0.960i)8-s + (0.692 + 0.721i)10-s + (0.964 + 0.265i)11-s + (−0.999 + 0.0268i)14-s + (−0.147 + 0.989i)16-s + (−0.970 − 0.239i)17-s + (−0.5 + 0.866i)19-s + (0.329 + 0.944i)20-s + (0.766 + 0.642i)22-s + (0.766 + 0.642i)23-s + (0.730 + 0.682i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $-0.963 + 0.267i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ -0.963 + 0.267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3915625291 + 2.872891077i\)
\(L(\frac12)\) \(\approx\) \(0.3915625291 + 2.872891077i\)
\(L(1)\) \(\approx\) \(1.498892413 + 1.064403070i\)
\(L(1)\) \(\approx\) \(1.498892413 + 1.064403070i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.909 + 0.416i)T \)
5 \( 1 + (0.930 + 0.367i)T \)
7 \( 1 + (-0.897 + 0.440i)T \)
11 \( 1 + (0.964 + 0.265i)T \)
17 \( 1 + (-0.970 - 0.239i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.909 + 0.416i)T \)
31 \( 1 + (-0.956 - 0.291i)T \)
37 \( 1 + (-0.919 + 0.391i)T \)
41 \( 1 + (-0.673 + 0.739i)T \)
43 \( 1 + (-0.956 + 0.291i)T \)
47 \( 1 + (-0.982 - 0.186i)T \)
53 \( 1 + (-0.970 - 0.239i)T \)
59 \( 1 + (-0.147 - 0.989i)T \)
61 \( 1 + (-0.589 + 0.807i)T \)
67 \( 1 + (0.0134 - 0.999i)T \)
71 \( 1 + (0.885 - 0.464i)T \)
73 \( 1 + (0.428 - 0.903i)T \)
79 \( 1 + (0.329 + 0.944i)T \)
83 \( 1 + (0.977 - 0.213i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.783 - 0.621i)T \)
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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

−17.802352703589077021120645002280, −17.148412287509661068147917473220, −16.553015320405361276742792699859, −15.85594765482253458683604610982, −15.115802477905398801770256163028, −14.34692383087047866716844239020, −13.7088246743482651889194722993, −13.25391870361886992565035843008, −12.655910662587799970530253978257, −12.050850468427255433667963877065, −11.01942981989209190396800163458, −10.610085653389331575877672665258, −9.80212681953253858584871527479, −9.1470993341734719809159588363, −8.58553137966575426153383091761, −7.02299412017999514508731276984, −6.62677682973136575155806191086, −6.173932331590495700405426595846, −5.18820581001981155116020224913, −4.57237903458349381380779570237, −3.78660137121612396290574641318, −3.025659519069446517679639883739, −2.21937028565614247898669423317, −1.45163764333316639289631411649, −0.482688156795787171606693086281, 1.60854013731890879164842785273, 2.11938831974475541451104739583, 3.22855329111753290394842688926, 3.483412810296602077148106287351, 4.698549461390422868052411140, 5.25167679571666427470520076584, 6.28103025478131770615861443272, 6.48133828128748111148735542074, 7.05540057107885166304095288968, 8.172879379759058119178171353094, 9.02295066842746031057181241238, 9.55378088120793975073244309565, 10.42855227623668902127553708868, 11.19783374860514019910109926163, 11.956024615421066391632307046264, 12.64931700577084428502750986373, 13.21301779800516580046941376001, 13.8019301111206364033641747870, 14.50973646209049014818231770463, 15.090279289064653279940337253178, 15.66601548783961709036144589359, 16.68562829917303689639252292213, 16.86114149199650849503293780533, 17.78734296871554187416198833857, 18.35610648460465147607705851275

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