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Properties

Label 1-569-569.104-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.582 - 0.812i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (0.759 − 0.650i)2-s + (0.0663 − 0.997i)3-s + (0.154 − 0.988i)4-s + (0.699 + 0.714i)5-s + (−0.598 − 0.801i)6-s + (0.562 − 0.826i)7-s + (−0.525 − 0.850i)8-s + (−0.991 − 0.132i)9-s + (0.996 + 0.0883i)10-s + (0.862 + 0.506i)11-s + (−0.975 − 0.219i)12-s + (0.814 − 0.580i)13-s + (−0.110 − 0.993i)14-s + (0.759 − 0.650i)15-s + (−0.952 − 0.304i)16-s + (0.408 − 0.912i)17-s + ⋯
L(s)  = 1  + (0.759 − 0.650i)2-s + (0.0663 − 0.997i)3-s + (0.154 − 0.988i)4-s + (0.699 + 0.714i)5-s + (−0.598 − 0.801i)6-s + (0.562 − 0.826i)7-s + (−0.525 − 0.850i)8-s + (−0.991 − 0.132i)9-s + (0.996 + 0.0883i)10-s + (0.862 + 0.506i)11-s + (−0.975 − 0.219i)12-s + (0.814 − 0.580i)13-s + (−0.110 − 0.993i)14-s + (0.759 − 0.650i)15-s + (−0.952 − 0.304i)16-s + (0.408 − 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.582 - 0.812i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.582 - 0.812i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.198317488 - 2.334064552i\)
\(L(\frac12)\) \(\approx\) \(1.198317488 - 2.334064552i\)
\(L(1)\) \(\approx\) \(1.403646343 - 1.269052434i\)
\(L(1)\) \(\approx\) \(1.403646343 - 1.269052434i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.759 - 0.650i)T \)
3 \( 1 + (0.0663 - 0.997i)T \)
5 \( 1 + (0.699 + 0.714i)T \)
7 \( 1 + (0.562 - 0.826i)T \)
11 \( 1 + (0.862 + 0.506i)T \)
13 \( 1 + (0.814 - 0.580i)T \)
17 \( 1 + (0.408 - 0.912i)T \)
19 \( 1 + (0.154 + 0.988i)T \)
23 \( 1 + (-0.787 + 0.616i)T \)
29 \( 1 + (0.408 + 0.912i)T \)
31 \( 1 + (-0.448 - 0.894i)T \)
37 \( 1 + (0.562 - 0.826i)T \)
41 \( 1 + (-0.787 + 0.616i)T \)
43 \( 1 + (0.759 + 0.650i)T \)
47 \( 1 + (-0.839 - 0.544i)T \)
53 \( 1 + (-0.598 - 0.801i)T \)
59 \( 1 + (-0.839 - 0.544i)T \)
61 \( 1 + (-0.730 + 0.683i)T \)
67 \( 1 + (0.408 + 0.912i)T \)
71 \( 1 + (-0.525 + 0.850i)T \)
73 \( 1 + (0.154 + 0.988i)T \)
79 \( 1 + (-0.283 - 0.958i)T \)
83 \( 1 + (-0.367 - 0.930i)T \)
89 \( 1 + (-0.448 + 0.894i)T \)
97 \( 1 + (-0.0221 + 0.999i)T \)
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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

−23.75098006305445119315277963334, −22.44433874945457959268484573837, −21.78321687319657420803891593555, −21.34928148595799382316454424757, −20.67965667711796742962922015196, −19.69890008600586708472973164282, −18.23880200627440701392421307047, −17.25252169711457668518728477784, −16.74127530574917633135979208391, −15.86536551706921684176543037226, −15.19982366335697967235475764233, −14.1399041141986974061283144003, −13.81437194239022778802271488088, −12.477819207852665237668448278859, −11.72566418283140339694462533458, −10.82234072854480676675765356259, −9.394931705161073319776946704956, −8.7333139215969007918792388727, −8.18364878348241060537898638475, −6.30619461752788102088556023154, −5.8904726615216295108404530011, −4.848421575899352423683091405599, −4.19743452506276717638409291227, −3.05840373456440932077234682051, −1.79231170452532067983117813358, 1.20566972987682916641089468329, 1.80477142974459620701959817713, 3.037712621799085739035078439066, 3.90444713983893873827018531984, 5.34449593608286781577577947275, 6.1517893618656313248719911304, 6.98383420230375098803992382366, 7.87075152026931264099194370386, 9.374557157710554175360335698457, 10.237175665681381009384909594421, 11.20322643381142867618788969593, 11.80847934191482940255369497732, 12.91420715477983693808733536015, 13.600426328672779088913298006029, 14.39976746859152761767327491517, 14.63874659207500280539593456268, 16.23997003783760777221868793483, 17.47131404972447730229906604594, 18.1443234009944894993125589378, 18.735569785696095371512373890756, 19.92856599957833215394997154478, 20.34312991997803477370709955017, 21.25612372190293096855396211511, 22.3445938566612152558067388633, 23.00187838052573152845583816720

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