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Properties

Label 1-569-569.63-r0-0-0
Degree $1$
Conductor $569$
Sign $0.287 + 0.957i$
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  + (0.699 + 0.714i)2-s + (−0.787 + 0.616i)3-s + (−0.0221 + 0.999i)4-s + (−0.110 − 0.993i)5-s + (−0.991 − 0.132i)6-s + (−0.952 − 0.304i)7-s + (−0.730 + 0.683i)8-s + (0.240 − 0.970i)9-s + (0.633 − 0.773i)10-s + (0.562 + 0.826i)11-s + (−0.598 − 0.801i)12-s + (0.996 − 0.0883i)13-s + (−0.448 − 0.894i)14-s + (0.699 + 0.714i)15-s + (−0.999 − 0.0442i)16-s + (0.487 − 0.873i)17-s + ⋯
L(s)  = 1  + (0.699 + 0.714i)2-s + (−0.787 + 0.616i)3-s + (−0.0221 + 0.999i)4-s + (−0.110 − 0.993i)5-s + (−0.991 − 0.132i)6-s + (−0.952 − 0.304i)7-s + (−0.730 + 0.683i)8-s + (0.240 − 0.970i)9-s + (0.633 − 0.773i)10-s + (0.562 + 0.826i)11-s + (−0.598 − 0.801i)12-s + (0.996 − 0.0883i)13-s + (−0.448 − 0.894i)14-s + (0.699 + 0.714i)15-s + (−0.999 − 0.0442i)16-s + (0.487 − 0.873i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.287 + 0.957i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.287 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.116409837 + 0.8302311409i\)
\(L(\frac12)\) \(\approx\) \(1.116409837 + 0.8302311409i\)
\(L(1)\) \(\approx\) \(1.014955179 + 0.5126393731i\)
\(L(1)\) \(\approx\) \(1.014955179 + 0.5126393731i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.699 + 0.714i)T \)
3 \( 1 + (-0.787 + 0.616i)T \)
5 \( 1 + (-0.110 - 0.993i)T \)
7 \( 1 + (-0.952 - 0.304i)T \)
11 \( 1 + (0.562 + 0.826i)T \)
13 \( 1 + (0.996 - 0.0883i)T \)
17 \( 1 + (0.487 - 0.873i)T \)
19 \( 1 + (-0.0221 - 0.999i)T \)
23 \( 1 + (0.937 + 0.346i)T \)
29 \( 1 + (0.487 + 0.873i)T \)
31 \( 1 + (0.0663 + 0.997i)T \)
37 \( 1 + (-0.952 - 0.304i)T \)
41 \( 1 + (0.937 + 0.346i)T \)
43 \( 1 + (0.699 - 0.714i)T \)
47 \( 1 + (0.862 + 0.506i)T \)
53 \( 1 + (-0.991 - 0.132i)T \)
59 \( 1 + (0.862 + 0.506i)T \)
61 \( 1 + (0.325 + 0.945i)T \)
67 \( 1 + (0.487 + 0.873i)T \)
71 \( 1 + (-0.730 - 0.683i)T \)
73 \( 1 + (-0.0221 - 0.999i)T \)
79 \( 1 + (0.964 - 0.262i)T \)
83 \( 1 + (0.814 + 0.580i)T \)
89 \( 1 + (0.0663 - 0.997i)T \)
97 \( 1 + (-0.975 + 0.219i)T \)
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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

−22.98253809740409931091401233718, −22.37989176522238260534076119335, −21.704186289628847449150061185325, −20.81383099360804221249223587971, −19.30840506095186741334219286826, −19.02976741059136820487208760753, −18.57226081070148053696792669073, −17.33563765241683852842501734816, −16.26043407549656889391411108128, −15.44314346723479079256760750358, −14.33712701038818452481835350119, −13.6106251953907403323504553589, −12.761588105999517117250170947364, −11.99659804534699641884629805714, −11.13447027421536961798065743813, −10.582072546556756751389361202670, −9.617394233924735422006789903242, −8.22725353279853665666910375493, −6.79899915658068256501943640218, −6.127698971498221922416208759206, −5.73483335166615112548148612215, −4.02582375341904603899082499022, −3.29570072728684627779650588136, −2.17918180972095663397094358736, −0.90074223363400761597764364389, 0.95352945424171013345736620741, 3.129320517034011902740518326015, 4.02897665336703422133096901843, 4.84344066810370428881543009464, 5.59008110762473890705616371711, 6.64299694972696621924639179069, 7.30138228687034929194210428823, 8.953822283403435778872756864970, 9.26130360334696977066525623352, 10.630187407545312354269936343607, 11.75048941144527009250335425859, 12.455539460776608289811306973611, 13.11919447072689712010591297005, 14.107090793093447885212363128582, 15.34220592161688463883779436247, 15.992768999579138088042667984784, 16.3918408151465730529679254202, 17.35199759380255625809888350966, 17.86410021987093045749193619834, 19.4434678853019684983277127627, 20.534743598767219852066953899843, 20.970120996488822652322892994211, 22.039042945682689286611884403167, 22.75110893044325185667945686592, 23.35459826214473640523244330269

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