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Properties

Label 1-569-569.25-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.993 - 0.111i$
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  + (−0.952 − 0.304i)2-s + (0.964 + 0.262i)3-s + (0.814 + 0.580i)4-s + (−0.999 − 0.0442i)5-s + (−0.839 − 0.544i)6-s + (−0.730 + 0.683i)7-s + (−0.598 − 0.801i)8-s + (0.862 + 0.506i)9-s + (0.937 + 0.346i)10-s + (−0.525 + 0.850i)11-s + (0.633 + 0.773i)12-s + (−0.787 − 0.616i)13-s + (0.903 − 0.428i)14-s + (−0.952 − 0.304i)15-s + (0.325 + 0.945i)16-s + (−0.110 + 0.993i)17-s + ⋯
L(s)  = 1  + (−0.952 − 0.304i)2-s + (0.964 + 0.262i)3-s + (0.814 + 0.580i)4-s + (−0.999 − 0.0442i)5-s + (−0.839 − 0.544i)6-s + (−0.730 + 0.683i)7-s + (−0.598 − 0.801i)8-s + (0.862 + 0.506i)9-s + (0.937 + 0.346i)10-s + (−0.525 + 0.850i)11-s + (0.633 + 0.773i)12-s + (−0.787 − 0.616i)13-s + (0.903 − 0.428i)14-s + (−0.952 − 0.304i)15-s + (0.325 + 0.945i)16-s + (−0.110 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.993 - 0.111i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.993 - 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005984044489 + 0.1068341230i\)
\(L(\frac12)\) \(\approx\) \(0.005984044489 + 0.1068341230i\)
\(L(1)\) \(\approx\) \(0.5660690969 + 0.06819522912i\)
\(L(1)\) \(\approx\) \(0.5660690969 + 0.06819522912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.952 - 0.304i)T \)
3 \( 1 + (0.964 + 0.262i)T \)
5 \( 1 + (-0.999 - 0.0442i)T \)
7 \( 1 + (-0.730 + 0.683i)T \)
11 \( 1 + (-0.525 + 0.850i)T \)
13 \( 1 + (-0.787 - 0.616i)T \)
17 \( 1 + (-0.110 + 0.993i)T \)
19 \( 1 + (0.814 - 0.580i)T \)
23 \( 1 + (-0.883 - 0.467i)T \)
29 \( 1 + (-0.110 - 0.993i)T \)
31 \( 1 + (-0.283 - 0.958i)T \)
37 \( 1 + (-0.730 + 0.683i)T \)
41 \( 1 + (-0.883 - 0.467i)T \)
43 \( 1 + (-0.952 + 0.304i)T \)
47 \( 1 + (-0.666 + 0.745i)T \)
53 \( 1 + (-0.839 - 0.544i)T \)
59 \( 1 + (-0.666 + 0.745i)T \)
61 \( 1 + (-0.991 - 0.132i)T \)
67 \( 1 + (-0.110 - 0.993i)T \)
71 \( 1 + (-0.598 + 0.801i)T \)
73 \( 1 + (0.814 - 0.580i)T \)
79 \( 1 + (0.408 - 0.912i)T \)
83 \( 1 + (0.0663 - 0.997i)T \)
89 \( 1 + (-0.283 + 0.958i)T \)
97 \( 1 + (0.996 + 0.0883i)T \)
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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

−23.239553883233212458966097497493, −21.89669332305802196402298806921, −20.70439684102342333028114163684, −19.89937477396526868059802396211, −19.624581459542314677384052985743, −18.66886440353689026228611807793, −18.2014092176116427325746263840, −16.6856631571923216154665187863, −16.10611507912743429707879290603, −15.531057696656625793048029415432, −14.34109050639168187938258279651, −13.8156599783296779195801266210, −12.45863499243111222541766106811, −11.63523819169069128026249169785, −10.50283301419781642114528978352, −9.68083922565441346800790946975, −8.84375856329990181248668229927, −7.91984705244656277554723707918, −7.282576321465828658770536830854, −6.657836806516993062634242161525, −5.07003735187272776178650689139, −3.55377485894553873961506786170, −2.94954193834134094876212623015, −1.4897414885999029917409790442, −0.06412710145403301178714718059, 1.92495225779994742879136810058, 2.83478103581628203125581497475, 3.602512576504883142750058502677, 4.80231242502425706825060486134, 6.47160150858301365753291019817, 7.61528692467719279918914681409, 8.00047352919176135648922182774, 8.99790724924054303374591761625, 9.838806110289696319328999656181, 10.43698719482466629846580514359, 11.77321486796830628238602346289, 12.49019220574859517979959111125, 13.215865962934627059901007871260, 14.93003684285024881066938078158, 15.36207928945828484818424178983, 15.92855799198317193021321503341, 16.96611451979997875878521649045, 18.144052270681409789054409694980, 18.87047129464268794973852837653, 19.61453649459640738008619060659, 20.08821320420817094126057521610, 20.81240570158218286866721548567, 21.95741949532391383951208375348, 22.55360405271189628142293017909, 24.09413013377137936884769020762

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