↑

Properties

Label 1-569-569.142-r0-0-0
Degree $1$
Conductor $569$
Sign $0.911 - 0.411i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.562 + 0.826i)2-s + (−0.283 − 0.958i)3-s + (−0.367 + 0.930i)4-s + (−0.952 + 0.304i)5-s + (0.633 − 0.773i)6-s + (−0.525 + 0.850i)7-s + (−0.975 + 0.219i)8-s + (−0.839 + 0.544i)9-s + (−0.787 − 0.616i)10-s + (−0.666 − 0.745i)11-s + (0.996 + 0.0883i)12-s + (0.0663 − 0.997i)13-s + (−0.999 + 0.0442i)14-s + (0.562 + 0.826i)15-s + (−0.730 − 0.683i)16-s + (0.699 + 0.714i)17-s + ⋯
L(s)  = 1  + (0.562 + 0.826i)2-s + (−0.283 − 0.958i)3-s + (−0.367 + 0.930i)4-s + (−0.952 + 0.304i)5-s + (0.633 − 0.773i)6-s + (−0.525 + 0.850i)7-s + (−0.975 + 0.219i)8-s + (−0.839 + 0.544i)9-s + (−0.787 − 0.616i)10-s + (−0.666 − 0.745i)11-s + (0.996 + 0.0883i)12-s + (0.0663 − 0.997i)13-s + (−0.999 + 0.0442i)14-s + (0.562 + 0.826i)15-s + (−0.730 − 0.683i)16-s + (0.699 + 0.714i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.911 - 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8348563627 - 0.1798699459i\)
\(L(\frac12)\) \(\approx\) \(0.8348563627 - 0.1798699459i\)
\(L(1)\) \(\approx\) \(0.8577450689 + 0.1522237621i\)
\(L(1)\) \(\approx\) \(0.8577450689 + 0.1522237621i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.562 + 0.826i)T \)
3 \( 1 + (-0.283 - 0.958i)T \)
5 \( 1 + (-0.952 + 0.304i)T \)
7 \( 1 + (-0.525 + 0.850i)T \)
11 \( 1 + (-0.666 - 0.745i)T \)
13 \( 1 + (0.0663 - 0.997i)T \)
17 \( 1 + (0.699 + 0.714i)T \)
19 \( 1 + (-0.367 - 0.930i)T \)
23 \( 1 + (0.964 - 0.262i)T \)
29 \( 1 + (0.699 - 0.714i)T \)
31 \( 1 + (0.903 + 0.428i)T \)
37 \( 1 + (-0.525 + 0.850i)T \)
41 \( 1 + (0.964 - 0.262i)T \)
43 \( 1 + (0.562 - 0.826i)T \)
47 \( 1 + (-0.921 + 0.387i)T \)
53 \( 1 + (0.633 - 0.773i)T \)
59 \( 1 + (-0.921 + 0.387i)T \)
61 \( 1 + (-0.598 + 0.801i)T \)
67 \( 1 + (0.699 - 0.714i)T \)
71 \( 1 + (-0.975 - 0.219i)T \)
73 \( 1 + (-0.367 - 0.930i)T \)
79 \( 1 + (-0.197 + 0.980i)T \)
83 \( 1 + (-0.448 - 0.894i)T \)
89 \( 1 + (0.903 - 0.428i)T \)
97 \( 1 + (0.814 - 0.580i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.13486120929955243927581144496, −22.77797746778906205834601648539, −21.39539089778729556841968563818, −20.902010636310134994008801377253, −20.19736578316266790501982772118, −19.43171610615009478167242293898, −18.62494341329345526149533682418, −17.33601902593879442434665091102, −16.31369183224474387848881098792, −15.806618113179794289373755645885, −14.762786417983139345026562967912, −14.080965521081308519902561627629, −12.86684338215199827438821198786, −12.136807879401795264238048669258, −11.30468450521330688581742093809, −10.51431445229884658199514651741, −9.768968509836076194726103308004, −8.94623151090929381637620472628, −7.56366703491740456367309559042, −6.40295798161383737813886535586, −5.091698916016540969157923302, −4.44173767869397981644059080280, −3.72022878064467550147465545876, −2.82211863999298998369682773959, −1.03233966435967264536709998617, 0.47088912496857656819734840130, 2.78566051163806612359518605721, 3.15336838312815700343938864112, 4.765388858090344414262304035956, 5.73364638767516137152339033686, 6.42121414898534835062730423362, 7.37061065058340399304021797508, 8.231521421776928096158350001167, 8.6922382113604918767276489348, 10.53295252665255960902061543810, 11.55921138190704569460336177827, 12.37939041472016095171713282314, 12.92898351815327491737315113444, 13.77530123481574988790790828357, 14.96303711095329409848332553821, 15.510434541750515231569837504028, 16.32054821507799115921832146564, 17.31992447280960500795246880072, 18.146096676450652464775404207891, 19.018070651879041120554234549466, 19.47302711043646361150689260365, 20.93399026683708830952202120076, 21.85994287211284097304806664784, 22.86964202675053411093521603893, 23.05944272230502408864087092726

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