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Properties

Label 1-569-569.215-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.450 - 0.892i$
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  + (−0.598 + 0.801i)2-s + (−0.699 + 0.714i)3-s + (−0.283 − 0.958i)4-s + (−0.991 + 0.132i)5-s + (−0.154 − 0.988i)6-s + (0.633 − 0.773i)7-s + (0.937 + 0.346i)8-s + (−0.0221 − 0.999i)9-s + (0.487 − 0.873i)10-s + (−0.996 + 0.0883i)11-s + (0.883 + 0.467i)12-s + (0.408 + 0.912i)13-s + (0.240 + 0.970i)14-s + (0.598 − 0.801i)15-s + (−0.839 + 0.544i)16-s + (0.325 + 0.945i)17-s + ⋯
L(s)  = 1  + (−0.598 + 0.801i)2-s + (−0.699 + 0.714i)3-s + (−0.283 − 0.958i)4-s + (−0.991 + 0.132i)5-s + (−0.154 − 0.988i)6-s + (0.633 − 0.773i)7-s + (0.937 + 0.346i)8-s + (−0.0221 − 0.999i)9-s + (0.487 − 0.873i)10-s + (−0.996 + 0.0883i)11-s + (0.883 + 0.467i)12-s + (0.408 + 0.912i)13-s + (0.240 + 0.970i)14-s + (0.598 − 0.801i)15-s + (−0.839 + 0.544i)16-s + (0.325 + 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.450 - 0.892i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.450 - 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02729148887 + 0.04434411047i\)
\(L(\frac12)\) \(\approx\) \(0.02729148887 + 0.04434411047i\)
\(L(1)\) \(\approx\) \(0.3893434631 + 0.2129317445i\)
\(L(1)\) \(\approx\) \(0.3893434631 + 0.2129317445i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.598 + 0.801i)T \)
3 \( 1 + (-0.699 + 0.714i)T \)
5 \( 1 + (-0.991 + 0.132i)T \)
7 \( 1 + (0.633 - 0.773i)T \)
11 \( 1 + (-0.996 + 0.0883i)T \)
13 \( 1 + (0.408 + 0.912i)T \)
17 \( 1 + (0.325 + 0.945i)T \)
19 \( 1 + (0.283 - 0.958i)T \)
23 \( 1 + (0.110 - 0.993i)T \)
29 \( 1 + (-0.325 + 0.945i)T \)
31 \( 1 + (-0.759 + 0.650i)T \)
37 \( 1 + (-0.633 + 0.773i)T \)
41 \( 1 + (-0.110 + 0.993i)T \)
43 \( 1 + (-0.598 - 0.801i)T \)
47 \( 1 + (-0.814 + 0.580i)T \)
53 \( 1 + (-0.154 - 0.988i)T \)
59 \( 1 + (-0.814 + 0.580i)T \)
61 \( 1 + (-0.921 + 0.387i)T \)
67 \( 1 + (0.325 - 0.945i)T \)
71 \( 1 + (0.937 - 0.346i)T \)
73 \( 1 + (0.283 - 0.958i)T \)
79 \( 1 + (-0.952 - 0.304i)T \)
83 \( 1 + (0.197 + 0.980i)T \)
89 \( 1 + (-0.759 - 0.650i)T \)
97 \( 1 + (-0.964 + 0.262i)T \)
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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

−22.93136537234090368855640744712, −21.79944310923475051761348543207, −20.854005277342306193099418937932, −20.189553143891484729823888681724, −19.08590354642011662172816231203, −18.50719354894049792632758565340, −18.05552581785408184811253281239, −17.05924989434626016795709950331, −16.07553238632335958997165836752, −15.42848870232867884311233101998, −13.89819580772392740271125462039, −12.86690040972811079711032952511, −12.29244289441231059664069705477, −11.44094542450826322990253738883, −11.030012875685588121127575718, −9.900775461295970506536520651107, −8.5772980935441726519195702581, −7.787726782139841578637969660212, −7.46037013257229274401017604600, −5.680681445096082189144604099014, −4.960598893294223838047743269318, −3.55830241060109665085206708384, −2.487004554782487969009211364768, −1.32957399907606933670235844521, −0.04147894463691430980219586929, 1.38913551357509348498121890427, 3.4822408832189307342084888619, 4.60526926863171224542704854978, 5.04482987063925744163177288508, 6.4603711383699679267078235749, 7.16596071780977263701123209483, 8.16696823199167686096380434022, 8.94067868523563307390665120555, 10.226588694189422461776665081880, 10.82455729255273488722857909300, 11.40509264668089482938729679730, 12.7162016998425162988994364864, 14.02189623341674359583675682849, 14.917978071198039053450339130950, 15.49537608747230683904140494492, 16.48992589050652641953582557601, 16.7499075815286958646858791434, 17.97428689849569062078771467689, 18.45881343124244384043693792211, 19.620347369450356949037753358895, 20.404818772657794241733351582247, 21.32438747141209892257399546457, 22.442026321472667137753226688405, 23.325061774546867178732112617764, 23.82457149805624900227967020183

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