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Properties

Label 1-569-569.66-r0-0-0
Degree $1$
Conductor $569$
Sign $0.689 - 0.724i$
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  + (0.0221 − 0.999i)2-s + (0.970 + 0.240i)3-s + (−0.999 − 0.0442i)4-s + (−0.975 + 0.219i)5-s + (0.262 − 0.964i)6-s + (−0.814 − 0.580i)7-s + (−0.0663 + 0.997i)8-s + (0.883 + 0.467i)9-s + (0.197 + 0.980i)10-s + (0.930 + 0.367i)11-s + (−0.958 − 0.283i)12-s + (−0.984 + 0.176i)13-s + (−0.598 + 0.801i)14-s + (−0.999 − 0.0221i)15-s + (0.996 + 0.0883i)16-s + (0.525 + 0.850i)17-s + ⋯
L(s)  = 1  + (0.0221 − 0.999i)2-s + (0.970 + 0.240i)3-s + (−0.999 − 0.0442i)4-s + (−0.975 + 0.219i)5-s + (0.262 − 0.964i)6-s + (−0.814 − 0.580i)7-s + (−0.0663 + 0.997i)8-s + (0.883 + 0.467i)9-s + (0.197 + 0.980i)10-s + (0.930 + 0.367i)11-s + (−0.958 − 0.283i)12-s + (−0.984 + 0.176i)13-s + (−0.598 + 0.801i)14-s + (−0.999 − 0.0221i)15-s + (0.996 + 0.0883i)16-s + (0.525 + 0.850i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.689 - 0.724i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.689 - 0.724i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.279799941 - 0.5488005480i\)
\(L(\frac12)\) \(\approx\) \(1.279799941 - 0.5488005480i\)
\(L(1)\) \(\approx\) \(1.055567274 - 0.3929505323i\)
\(L(1)\) \(\approx\) \(1.055567274 - 0.3929505323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.0221 - 0.999i)T \)
3 \( 1 + (0.970 + 0.240i)T \)
5 \( 1 + (-0.975 + 0.219i)T \)
7 \( 1 + (-0.814 - 0.580i)T \)
11 \( 1 + (0.930 + 0.367i)T \)
13 \( 1 + (-0.984 + 0.176i)T \)
17 \( 1 + (0.525 + 0.850i)T \)
19 \( 1 + (-0.0442 - 0.999i)T \)
23 \( 1 + (0.650 - 0.759i)T \)
29 \( 1 + (0.850 + 0.525i)T \)
31 \( 1 + (0.132 + 0.991i)T \)
37 \( 1 + (0.580 - 0.814i)T \)
41 \( 1 + (0.759 + 0.650i)T \)
43 \( 1 + (-0.0221 - 0.999i)T \)
47 \( 1 + (0.873 - 0.487i)T \)
53 \( 1 + (-0.262 + 0.964i)T \)
59 \( 1 + (-0.873 + 0.487i)T \)
61 \( 1 + (0.787 - 0.616i)T \)
67 \( 1 + (0.525 - 0.850i)T \)
71 \( 1 + (-0.0663 - 0.997i)T \)
73 \( 1 + (0.0442 + 0.999i)T \)
79 \( 1 + (-0.862 + 0.506i)T \)
83 \( 1 + (0.945 - 0.325i)T \)
89 \( 1 + (-0.132 + 0.991i)T \)
97 \( 1 + (-0.428 - 0.903i)T \)
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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

−23.48549713620812910804987549055, −22.702950260614841401173529693544, −21.95359853551555775851151289395, −20.82950390253961486891420548770, −19.68754257881628412129298550370, −19.11558669517354769746712366838, −18.66219355475233794992879913608, −17.331477899071270050164784833025, −16.37353539775959784394188009802, −15.74670169077954617185840467847, −14.90799997412445001280348197424, −14.3537719699813013636662193889, −13.30433329717539408495298065816, −12.45243387096909486602373138597, −11.79053345341004584822062183770, −9.825883949008167684951561113789, −9.35867987358768375831254803908, −8.40995722114797582118439315007, −7.66209208399177991220760633210, −6.92403641504776969776603807998, −5.88755187300679445286257178796, −4.592485359036611114168879323068, −3.66505293518057513064480707851, −2.84750363794832085662041055131, −0.9037619910959625680243910245, 0.99231635916483805242347196550, 2.49054684190155532791664998237, 3.2861383265009283761907080842, 4.09970502434222605435118774848, 4.74575096223231210660784502875, 6.74026503777209742684242258153, 7.54033856684528189260358226790, 8.66797030423151497016939461622, 9.34765001584660052319032033054, 10.27582417366783602430834086993, 10.95362196472994442605507973216, 12.34606708540082099653656451190, 12.60482481990211340606933812749, 13.880381286720430477411315351400, 14.54144431848105739281721631785, 15.28597831899348521202338357302, 16.45845964148427533394410331926, 17.32893676520236563537646397774, 18.65602034097399705802826944702, 19.442770263879490258062509997304, 19.74260939596347754928162892460, 20.27429363119601628871486699553, 21.546615512216888846202860452614, 22.078264751452989472659579186149, 23.026909929186764701020497395408

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