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Properties

Label 1-569-569.153-r0-0-0
Degree $1$
Conductor $569$
Sign $0.00280 + 0.999i$
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  + (−0.110 − 0.993i)2-s + (0.937 + 0.346i)3-s + (−0.975 + 0.219i)4-s + (−0.448 − 0.894i)5-s + (0.240 − 0.970i)6-s + (−0.999 − 0.0442i)7-s + (0.325 + 0.945i)8-s + (0.759 + 0.650i)9-s + (−0.839 + 0.544i)10-s + (−0.952 + 0.304i)11-s + (−0.991 − 0.132i)12-s + (0.633 + 0.773i)13-s + (0.0663 + 0.997i)14-s + (−0.110 − 0.993i)15-s + (0.903 − 0.428i)16-s + (−0.367 − 0.930i)17-s + ⋯
L(s)  = 1  + (−0.110 − 0.993i)2-s + (0.937 + 0.346i)3-s + (−0.975 + 0.219i)4-s + (−0.448 − 0.894i)5-s + (0.240 − 0.970i)6-s + (−0.999 − 0.0442i)7-s + (0.325 + 0.945i)8-s + (0.759 + 0.650i)9-s + (−0.839 + 0.544i)10-s + (−0.952 + 0.304i)11-s + (−0.991 − 0.132i)12-s + (0.633 + 0.773i)13-s + (0.0663 + 0.997i)14-s + (−0.110 − 0.993i)15-s + (0.903 − 0.428i)16-s + (−0.367 − 0.930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00280 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00280 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.00280 + 0.999i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.00280 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1968699489 + 0.1963189302i\)
\(L(\frac12)\) \(\approx\) \(0.1968699489 + 0.1963189302i\)
\(L(1)\) \(\approx\) \(0.7121676109 - 0.2547790669i\)
\(L(1)\) \(\approx\) \(0.7121676109 - 0.2547790669i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.110 - 0.993i)T \)
3 \( 1 + (0.937 + 0.346i)T \)
5 \( 1 + (-0.448 - 0.894i)T \)
7 \( 1 + (-0.999 - 0.0442i)T \)
11 \( 1 + (-0.952 + 0.304i)T \)
13 \( 1 + (0.633 + 0.773i)T \)
17 \( 1 + (-0.367 - 0.930i)T \)
19 \( 1 + (-0.975 - 0.219i)T \)
23 \( 1 + (-0.921 + 0.387i)T \)
29 \( 1 + (-0.367 + 0.930i)T \)
31 \( 1 + (-0.787 - 0.616i)T \)
37 \( 1 + (-0.999 - 0.0442i)T \)
41 \( 1 + (-0.921 + 0.387i)T \)
43 \( 1 + (-0.110 + 0.993i)T \)
47 \( 1 + (0.562 + 0.826i)T \)
53 \( 1 + (0.240 - 0.970i)T \)
59 \( 1 + (0.562 + 0.826i)T \)
61 \( 1 + (0.984 + 0.176i)T \)
67 \( 1 + (-0.367 + 0.930i)T \)
71 \( 1 + (0.325 - 0.945i)T \)
73 \( 1 + (-0.975 - 0.219i)T \)
79 \( 1 + (-0.883 + 0.467i)T \)
83 \( 1 + (0.996 + 0.0883i)T \)
89 \( 1 + (-0.787 + 0.616i)T \)
97 \( 1 + (-0.598 + 0.801i)T \)
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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

−23.43493676289749955742071150850, −22.37211742790236537378820124598, −21.636634944044987474684381568721, −20.34611897345605496238259820043, −19.391922820152917034112490683231, −18.77380050205575303906800569002, −18.289240939940994138317516460895, −17.19919123141386347584451306595, −15.845308043837847283301682562842, −15.54001095805611683124613290820, −14.78413115792147416422359885243, −13.78818210011937156124153091093, −13.13645731511610536005587112426, −12.38559218347062358311316464301, −10.489552447038648533290679525733, −10.16332806796601032110685134636, −8.71588128947095526521841622476, −8.23302404267446069451566348583, −7.29403420549224292677785824656, −6.50084781204006817181625122865, −5.73136361686144465066391104466, −3.95087202246881601859819993891, −3.45258107475229594070118328389, −2.200029649797667284136057353617, −0.12610750975048350004124519766, 1.64004290266023291051752382228, 2.648570352716259835231838763371, 3.69428028565948171682945012668, 4.35107238885546470392286943793, 5.365063697559812981715626968251, 7.11842467738197359021561329208, 8.1895745989563389501624426578, 8.9479848793703722152386915063, 9.54953136172460263235229079172, 10.426999896219786129656120855773, 11.466487426739453530817272716665, 12.580665351072823675066574286230, 13.1814122595507736105032776436, 13.74968944671805689451101910391, 15.01080506916059773460410315742, 16.07014379026384958316798204244, 16.465704960456865428015313881739, 17.91646383860180320568788396933, 18.86696103760157179088100551861, 19.431886811903951802991881514472, 20.28693119499738725949031266119, 20.69329324755545212918021711934, 21.538106366832743378664301986883, 22.39953573835228897142777739856, 23.453032910164198784705824880012

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