Properties

Label 1-569-569.125-r0-0-0
Degree $1$
Conductor $569$
Sign $0.0537 + 0.998i$
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  + (−0.448 + 0.894i)2-s + (−0.921 − 0.387i)3-s + (−0.598 − 0.801i)4-s + (0.0663 − 0.997i)5-s + (0.759 − 0.650i)6-s + (0.903 + 0.428i)7-s + (0.984 − 0.176i)8-s + (0.699 + 0.714i)9-s + (0.862 + 0.506i)10-s + (−0.999 − 0.0442i)11-s + (0.240 + 0.970i)12-s + (−0.839 + 0.544i)13-s + (−0.787 + 0.616i)14-s + (−0.448 + 0.894i)15-s + (−0.283 + 0.958i)16-s + (0.814 − 0.580i)17-s + ⋯
L(s)  = 1  + (−0.448 + 0.894i)2-s + (−0.921 − 0.387i)3-s + (−0.598 − 0.801i)4-s + (0.0663 − 0.997i)5-s + (0.759 − 0.650i)6-s + (0.903 + 0.428i)7-s + (0.984 − 0.176i)8-s + (0.699 + 0.714i)9-s + (0.862 + 0.506i)10-s + (−0.999 − 0.0442i)11-s + (0.240 + 0.970i)12-s + (−0.839 + 0.544i)13-s + (−0.787 + 0.616i)14-s + (−0.448 + 0.894i)15-s + (−0.283 + 0.958i)16-s + (0.814 − 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0537 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0537 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.0537 + 0.998i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.0537 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4326218145 + 0.4099659073i\)
\(L(\frac12)\) \(\approx\) \(0.4326218145 + 0.4099659073i\)
\(L(1)\) \(\approx\) \(0.5847628937 + 0.1683441181i\)
\(L(1)\) \(\approx\) \(0.5847628937 + 0.1683441181i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.448 + 0.894i)T \)
3 \( 1 + (-0.921 - 0.387i)T \)
5 \( 1 + (0.0663 - 0.997i)T \)
7 \( 1 + (0.903 + 0.428i)T \)
11 \( 1 + (-0.999 - 0.0442i)T \)
13 \( 1 + (-0.839 + 0.544i)T \)
17 \( 1 + (0.814 - 0.580i)T \)
19 \( 1 + (-0.598 + 0.801i)T \)
23 \( 1 + (-0.666 + 0.745i)T \)
29 \( 1 + (0.814 + 0.580i)T \)
31 \( 1 + (0.937 + 0.346i)T \)
37 \( 1 + (0.903 + 0.428i)T \)
41 \( 1 + (-0.666 + 0.745i)T \)
43 \( 1 + (-0.448 - 0.894i)T \)
47 \( 1 + (-0.952 - 0.304i)T \)
53 \( 1 + (0.759 - 0.650i)T \)
59 \( 1 + (-0.952 - 0.304i)T \)
61 \( 1 + (-0.197 + 0.980i)T \)
67 \( 1 + (0.814 + 0.580i)T \)
71 \( 1 + (0.984 + 0.176i)T \)
73 \( 1 + (-0.598 + 0.801i)T \)
79 \( 1 + (0.154 + 0.988i)T \)
83 \( 1 + (0.633 + 0.773i)T \)
89 \( 1 + (0.937 - 0.346i)T \)
97 \( 1 + (-0.991 - 0.132i)T \)
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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

−23.045834531296577152105628369156, −21.944508666029970804636178136994, −21.47937498684653378110520011550, −20.7644619801221235536037950127, −19.70509879552760907155989154807, −18.71884700955682397519592494011, −17.98127416144991194696478831277, −17.47706523484645727856066806779, −16.71768621914687228238809766327, −15.459748959814983665491611493033, −14.64589817207247467819490184503, −13.55204477723385665454075422153, −12.49827211754978220360241738217, −11.71003090949786954867045425361, −10.80549793225374010097161411846, −10.39562024673463292675924537624, −9.754565855821054488121411378848, −8.13785135254841955786957738875, −7.5376428068273644791361834786, −6.302419176988919369956911729631, −5.01485708271747916716603531083, −4.3030604197093133784267313484, −3.04811797471700861121106352159, −2.04573730291751379811292142825, −0.486106921438587042800515019911, 1.07337998866384964940401922352, 2.03866434588933205561835879465, 4.4554409474270056113448202265, 5.136041385839804787355530897304, 5.63010273728507545890037353650, 6.808572436731710216822184421638, 7.94934090460488358454123922165, 8.25361522426475895991626014439, 9.66335384675499545848745342826, 10.32800698025369215143565779514, 11.63730513612199143820809581019, 12.287507765353141013497837367905, 13.33021108422531699389255172036, 14.18456106073819317169592558874, 15.28777896728221798504326670800, 16.17866435046403801893952507787, 16.75391430986400023997026886517, 17.51284473445027132401682146109, 18.23288541983015431071835467811, 18.89875259920786906134957805321, 19.95993814462210149580023653948, 21.21668623794632931598120975936, 21.751786652084233675391049380069, 23.164549923831920707387062082905, 23.584563800107075448426390123468

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