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Properties

Label 1-569-569.437-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: $\chi_{569} (437, \cdot )$
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

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

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