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Properties

Label 1-569-569.70-r0-0-0
Degree $1$
Conductor $569$
Sign $0.0134 - 0.999i$
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  + (0.984 + 0.176i)2-s + (0.367 − 0.930i)3-s + (0.937 + 0.346i)4-s + (−0.197 − 0.980i)5-s + (0.525 − 0.850i)6-s + (0.240 − 0.970i)7-s + (0.862 + 0.506i)8-s + (−0.730 − 0.683i)9-s + (−0.0221 − 0.999i)10-s + (0.991 − 0.132i)11-s + (0.666 − 0.745i)12-s + (0.154 − 0.988i)13-s + (0.408 − 0.912i)14-s + (−0.984 − 0.176i)15-s + (0.759 + 0.650i)16-s + (−0.283 + 0.958i)17-s + ⋯
L(s)  = 1  + (0.984 + 0.176i)2-s + (0.367 − 0.930i)3-s + (0.937 + 0.346i)4-s + (−0.197 − 0.980i)5-s + (0.525 − 0.850i)6-s + (0.240 − 0.970i)7-s + (0.862 + 0.506i)8-s + (−0.730 − 0.683i)9-s + (−0.0221 − 0.999i)10-s + (0.991 − 0.132i)11-s + (0.666 − 0.745i)12-s + (0.154 − 0.988i)13-s + (0.408 − 0.912i)14-s + (−0.984 − 0.176i)15-s + (0.759 + 0.650i)16-s + (−0.283 + 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0134 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0134 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.0134 - 0.999i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.0134 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.058060630 - 2.085959772i\)
\(L(\frac12)\) \(\approx\) \(2.058060630 - 2.085959772i\)
\(L(1)\) \(\approx\) \(1.895276480 - 0.9252961087i\)
\(L(1)\) \(\approx\) \(1.895276480 - 0.9252961087i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.984 + 0.176i)T \)
3 \( 1 + (0.367 - 0.930i)T \)
5 \( 1 + (-0.197 - 0.980i)T \)
7 \( 1 + (0.240 - 0.970i)T \)
11 \( 1 + (0.991 - 0.132i)T \)
13 \( 1 + (0.154 - 0.988i)T \)
17 \( 1 + (-0.283 + 0.958i)T \)
19 \( 1 + (-0.937 + 0.346i)T \)
23 \( 1 + (-0.814 + 0.580i)T \)
29 \( 1 + (0.283 + 0.958i)T \)
31 \( 1 + (-0.487 + 0.873i)T \)
37 \( 1 + (-0.240 + 0.970i)T \)
41 \( 1 + (0.814 - 0.580i)T \)
43 \( 1 + (0.984 - 0.176i)T \)
47 \( 1 + (0.598 - 0.801i)T \)
53 \( 1 + (0.525 - 0.850i)T \)
59 \( 1 + (0.598 - 0.801i)T \)
61 \( 1 + (0.562 + 0.826i)T \)
67 \( 1 + (-0.283 - 0.958i)T \)
71 \( 1 + (0.862 - 0.506i)T \)
73 \( 1 + (-0.937 + 0.346i)T \)
79 \( 1 + (-0.448 + 0.894i)T \)
83 \( 1 + (0.883 + 0.467i)T \)
89 \( 1 + (-0.487 - 0.873i)T \)
97 \( 1 + (0.921 - 0.387i)T \)
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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

−23.10192623073757119653831793124, −22.423744442227959897551199498461, −21.86144979219610021142212287007, −21.27020257374429521813385258746, −20.37314294522395048016467467170, −19.37544000299549463475944214858, −18.87673742909662145915437195052, −17.52604050595068419296136637666, −16.27095959360971970102834283656, −15.69560625427854134231701479857, −14.72863171891144391517592116706, −14.46639840408285268203820672330, −13.60315369059932779089781125898, −12.15991620954389588207982206072, −11.45495030941648175499966509874, −10.896817920225356510932177184485, −9.72236844145386121063879922628, −8.945169268275998513024377297519, −7.61404205883176996184532745565, −6.46992708645109783633824098850, −5.777376308983133529322587725973, −4.36006676146690664820824118517, −4.02689865777455658015697709569, −2.61970945443709124313228358707, −2.21165580572965269501356589961, 1.1125873742119118571306296818, 1.92669059547191505571475959502, 3.54619735049733676574072436895, 4.06434024183410239421449006482, 5.362900187952071373195365514551, 6.288124535107091316457421501578, 7.19144495922552957956407239091, 8.10521433135282952348842151226, 8.71472780698862282686641326082, 10.355581416482400154000732738247, 11.369553233611455783444841743590, 12.37691404052212234521611636209, 12.80544448015202247521905038368, 13.66589367196250991315753252536, 14.3559609830199333579889497047, 15.21737441036102703093916770151, 16.34313484600895403926756694650, 17.18032840376977446776776017930, 17.67470357979398616992774509051, 19.41872118997635557630846246018, 19.83140443900119917530642542470, 20.4421202496412271610942235493, 21.304626746652039945607545298086, 22.42647499830312862881803536325, 23.34735740805784702865540850338

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