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Properties

Label 1-569-569.65-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.475 + 0.879i$
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 + (−0.850 − 0.525i)3-s + (0.240 − 0.970i)4-s + (0.937 + 0.346i)5-s + (−0.993 + 0.110i)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 + (−0.958 + 0.283i)11-s + (−0.714 + 0.699i)12-s + (−0.562 + 0.826i)13-s + (−0.921 + 0.387i)14-s + (−0.616 − 0.787i)15-s + (−0.883 − 0.467i)16-s + (−0.633 − 0.773i)17-s + ⋯
L(s)  = 1  + (0.787 − 0.616i)2-s + (−0.850 − 0.525i)3-s + (0.240 − 0.970i)4-s + (0.937 + 0.346i)5-s + (−0.993 + 0.110i)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 + (−0.958 + 0.283i)11-s + (−0.714 + 0.699i)12-s + (−0.562 + 0.826i)13-s + (−0.921 + 0.387i)14-s + (−0.616 − 0.787i)15-s + (−0.883 − 0.467i)16-s + (−0.633 − 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.475 + 0.879i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.475 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1059806901 - 0.1776799859i\)
\(L(\frac12)\) \(\approx\) \(-0.1059806901 - 0.1776799859i\)
\(L(1)\) \(\approx\) \(0.7425310305 - 0.4539947289i\)
\(L(1)\) \(\approx\) \(0.7425310305 - 0.4539947289i\)

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

−23.86880969988774455115463932160, −22.84635527280935564399242271116, −22.04975389421180966512354581165, −21.78553908933051533523321006095, −20.834177529436950279488091053405, −20.00612228613114479004763506317, −18.403599455672960776562190639641, −17.73152256469502457168121379516, −16.71742051899556681160297926294, −16.46116553564485546522856383275, −15.31633653321144442292347169157, −14.85712760377456961539475080362, −13.277424729078128850166489521002, −12.96168630194349619403903549031, −12.23849745971083954996610356749, −10.88954406256993764778412038547, −10.16746493290196371849082502858, −9.1449264382481755524386863261, −8.08837647705054904619584280480, −6.63053935348790007861663193306, −6.10370138849061935495850207976, −5.33871735769337018438244859, −4.5477442366103545664667027417, −3.31466246616309740426442032935, −2.246611737318287439203397925692, 0.07921526768180735760447131265, 1.90258008661487086034002268325, 2.40684235219125725491286977858, 3.85805533438364847700846662266, 5.06562667128162000362662959746, 5.759778900242315794443025972204, 6.69686077387740360287003080859, 7.22895507353090507429981075132, 9.242504118756318197373739748223, 10.11699008188331364381440163763, 10.65180974831615994898980524981, 11.68868632832621029586301132223, 12.55939441005027238100110813044, 13.34728932985194564170483392299, 13.71195528396022272824172075201, 14.93975663530179325522708613296, 16.005223222198748819168689064353, 16.80073300285219836805697762598, 17.88914174003637513524725972651, 18.59626236397923019887970571669, 19.31285449381684678294924138755, 20.27171344428836438416897889677, 21.42460843474663718118342582732, 21.867470593648782964053746661558, 22.705034978722297949476433301581

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