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Properties

Label 1-569-569.24-r1-0-0
Degree $1$
Conductor $569$
Sign $0.316 + 0.948i$
Analytic cond. $61.1475$
Root an. cond. $61.1475$
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  + (−0.506 − 0.862i)2-s + (−0.336 − 0.941i)3-s + (−0.487 + 0.873i)4-s + (0.562 − 0.826i)5-s + (−0.641 + 0.766i)6-s + (−0.745 + 0.666i)7-s + (0.999 − 0.0221i)8-s + (−0.773 + 0.633i)9-s + (−0.997 − 0.0663i)10-s + (0.925 − 0.377i)11-s + (0.986 + 0.165i)12-s + (−0.894 + 0.448i)13-s + (0.952 + 0.304i)14-s + (−0.967 − 0.251i)15-s + (−0.525 − 0.850i)16-s + (0.650 − 0.759i)17-s + ⋯
L(s)  = 1  + (−0.506 − 0.862i)2-s + (−0.336 − 0.941i)3-s + (−0.487 + 0.873i)4-s + (0.562 − 0.826i)5-s + (−0.641 + 0.766i)6-s + (−0.745 + 0.666i)7-s + (0.999 − 0.0221i)8-s + (−0.773 + 0.633i)9-s + (−0.997 − 0.0663i)10-s + (0.925 − 0.377i)11-s + (0.986 + 0.165i)12-s + (−0.894 + 0.448i)13-s + (0.952 + 0.304i)14-s + (−0.967 − 0.251i)15-s + (−0.525 − 0.850i)16-s + (0.650 − 0.759i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ 0.316 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2565262951 - 0.1847849076i\)
\(L(\frac12)\) \(\approx\) \(-0.2565262951 - 0.1847849076i\)
\(L(1)\) \(\approx\) \(0.4256727396 - 0.4708904275i\)
\(L(1)\) \(\approx\) \(0.4256727396 - 0.4708904275i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.506 - 0.862i)T \)
3 \( 1 + (-0.336 - 0.941i)T \)
5 \( 1 + (0.562 - 0.826i)T \)
7 \( 1 + (-0.745 + 0.666i)T \)
11 \( 1 + (0.925 - 0.377i)T \)
13 \( 1 + (-0.894 + 0.448i)T \)
17 \( 1 + (0.650 - 0.759i)T \)
19 \( 1 + (0.273 - 0.962i)T \)
23 \( 1 + (-0.477 - 0.878i)T \)
29 \( 1 + (0.0773 - 0.997i)T \)
31 \( 1 + (-0.737 + 0.675i)T \)
37 \( 1 + (0.998 + 0.0552i)T \)
41 \( 1 + (0.283 - 0.958i)T \)
43 \( 1 + (-0.862 - 0.506i)T \)
47 \( 1 + (-0.418 + 0.908i)T \)
53 \( 1 + (0.766 + 0.641i)T \)
59 \( 1 + (-0.908 - 0.418i)T \)
61 \( 1 + (-0.219 + 0.975i)T \)
67 \( 1 + (0.650 + 0.759i)T \)
71 \( 1 + (-0.999 - 0.0221i)T \)
73 \( 1 + (-0.962 - 0.273i)T \)
79 \( 1 + (0.176 - 0.984i)T \)
83 \( 1 + (-0.624 - 0.780i)T \)
89 \( 1 + (-0.675 + 0.737i)T \)
97 \( 1 + (0.917 + 0.397i)T \)
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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

−23.452852306228042788564852668430, −22.93803387913798423713897699714, −22.1865871403560475412220142997, −21.602873634091765076797496790426, −20.06387059334569852209301023087, −19.662978153310834080760120394026, −18.43893207469552480114395143532, −17.62650882657309727486332140391, −16.79486914182990888410244488543, −16.50520134000559691150202003246, −15.18131498102755216656146183240, −14.682101400368648819363106799791, −14.029431368477958886399387161, −12.775759644059524281543422156, −11.419414561537448972036368353512, −10.347722438441682861181295102999, −9.863967483621155044173657340808, −9.39698544444286452797787918569, −7.92309673367351279676755544703, −6.96612893148398754105500886163, −6.13408763250832721048540222035, −5.436974636263373970433619643282, −4.132696290746451016731654494690, −3.2723975708885661890471177383, −1.487622397140654512973218345622, 0.11365628732685770163954950706, 1.01136852246051240752171861275, 2.139042113383376723768616610146, 2.90173942256901356848208183721, 4.46809028239725812697440667973, 5.53174608807467090387605974449, 6.57471722048057120671670596255, 7.57948798653040240258311533873, 8.7986872664108663020345658921, 9.23793706233647881529976238414, 10.18544480082791516364702883096, 11.619669494707064929226422374107, 12.02984553611009083566011048125, 12.752499276583612815200021290496, 13.55286662887091209285538535753, 14.3366130404591091647840215647, 16.19285755234345492417824547348, 16.73088183464000944150727245838, 17.48886489690533609543293454522, 18.30870873512055715587409437852, 19.12047134674410580438611289124, 19.72317791443106919612633605592, 20.47616191802394743641170534478, 21.78100945448122484268986057271, 22.05908361347507328120584852707

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