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Properties

Label 1-569-569.165-r0-0-0
Degree $1$
Conductor $569$
Sign $0.473 - 0.880i$
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  + (−0.787 − 0.616i)2-s + (−0.525 − 0.850i)3-s + (0.240 + 0.970i)4-s + (0.937 − 0.346i)5-s + (−0.110 + 0.993i)6-s + (0.964 − 0.262i)7-s + (0.408 − 0.912i)8-s + (−0.448 + 0.894i)9-s + (−0.952 − 0.304i)10-s + (−0.283 + 0.958i)11-s + (0.699 − 0.714i)12-s + (0.562 + 0.826i)13-s + (−0.921 − 0.387i)14-s + (−0.787 − 0.616i)15-s + (−0.883 + 0.467i)16-s + (0.633 − 0.773i)17-s + ⋯
L(s)  = 1  + (−0.787 − 0.616i)2-s + (−0.525 − 0.850i)3-s + (0.240 + 0.970i)4-s + (0.937 − 0.346i)5-s + (−0.110 + 0.993i)6-s + (0.964 − 0.262i)7-s + (0.408 − 0.912i)8-s + (−0.448 + 0.894i)9-s + (−0.952 − 0.304i)10-s + (−0.283 + 0.958i)11-s + (0.699 − 0.714i)12-s + (0.562 + 0.826i)13-s + (−0.921 − 0.387i)14-s + (−0.787 − 0.616i)15-s + (−0.883 + 0.467i)16-s + (0.633 − 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.473 - 0.880i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.473 - 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9183994705 - 0.5488580868i\)
\(L(\frac12)\) \(\approx\) \(0.9183994705 - 0.5488580868i\)
\(L(1)\) \(\approx\) \(0.7720463080 - 0.3630867913i\)
\(L(1)\) \(\approx\) \(0.7720463080 - 0.3630867913i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.787 - 0.616i)T \)
3 \( 1 + (-0.525 - 0.850i)T \)
5 \( 1 + (0.937 - 0.346i)T \)
7 \( 1 + (0.964 - 0.262i)T \)
11 \( 1 + (-0.283 + 0.958i)T \)
13 \( 1 + (0.562 + 0.826i)T \)
17 \( 1 + (0.633 - 0.773i)T \)
19 \( 1 + (0.240 - 0.970i)T \)
23 \( 1 + (-0.730 + 0.683i)T \)
29 \( 1 + (0.633 + 0.773i)T \)
31 \( 1 + (-0.666 + 0.745i)T \)
37 \( 1 + (0.964 - 0.262i)T \)
41 \( 1 + (-0.730 + 0.683i)T \)
43 \( 1 + (-0.787 + 0.616i)T \)
47 \( 1 + (0.903 + 0.428i)T \)
53 \( 1 + (-0.110 + 0.993i)T \)
59 \( 1 + (0.903 + 0.428i)T \)
61 \( 1 + (0.487 - 0.873i)T \)
67 \( 1 + (0.633 + 0.773i)T \)
71 \( 1 + (0.408 + 0.912i)T \)
73 \( 1 + (0.240 - 0.970i)T \)
79 \( 1 + (-0.975 + 0.219i)T \)
83 \( 1 + (0.862 - 0.506i)T \)
89 \( 1 + (-0.666 - 0.745i)T \)
97 \( 1 + (0.759 - 0.650i)T \)
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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

−23.5479381766031448904174871660, −22.55907082080485194453349289259, −21.6935464633805389522964113984, −20.89913444719172582084364554987, −20.309174777660010589760350339734, −18.70234246665861567691181111450, −18.300996568886744455321821705255, −17.4190980920458163981918611910, −16.815232541377134598524586084821, −15.99656661864064647921939812936, −15.001961931462633970637055348267, −14.48000132660324240518770698322, −13.54107979235818197111361564692, −11.96559734560991446811701369315, −10.94045809181399415329429857657, −10.41660211323648204354535235330, −9.73198016344086238587611554246, −8.52037399145879284663965854289, −8.04173767085454474158441047918, −6.42764899681776308032500890683, −5.687844153292246708513677165591, −5.31423905463340721070121442641, −3.76343426409587052588951998804, −2.28806348293778978217915316006, −0.979108870826752296665718292106, 1.151554810185972903048282745800, 1.73010827777185616761203875413, 2.699154462548188957397393335943, 4.48842357925215453348660531610, 5.35152691639257406254115532012, 6.71423484121584993491118257714, 7.39682865959522050093699018534, 8.35957783418679523897815931225, 9.31595215906139846370146644634, 10.23251504420328269278572202069, 11.19148535729923370745287943103, 11.84917387579254165885708423488, 12.74030653476369534632884581259, 13.57094656542771345899928388841, 14.29442939279083990857869790785, 15.98430104675606475902571025972, 16.77253509439781617596786696304, 17.609586813041268022770916481080, 18.04126231987381279688253826891, 18.58506515125440537009652014764, 19.938550983989076402657945764367, 20.39335529122121065317124265526, 21.46413958469202210154483720931, 21.92404809542421237992215686466, 23.30342909346556112827594819846

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