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Properties

Label 1-569-569.151-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.631 - 0.775i$
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  + (−0.0663 + 0.997i)2-s + (0.745 + 0.666i)3-s + (−0.991 − 0.132i)4-s + (−0.787 + 0.616i)5-s + (−0.714 + 0.699i)6-s + (0.283 − 0.958i)7-s + (0.197 − 0.980i)8-s + (0.110 + 0.993i)9-s + (−0.562 − 0.826i)10-s + (0.428 + 0.903i)11-s + (−0.650 − 0.759i)12-s + (−0.862 + 0.506i)13-s + (0.937 + 0.346i)14-s + (−0.997 − 0.0663i)15-s + (0.964 + 0.262i)16-s + (−0.996 + 0.0883i)17-s + ⋯
L(s)  = 1  + (−0.0663 + 0.997i)2-s + (0.745 + 0.666i)3-s + (−0.991 − 0.132i)4-s + (−0.787 + 0.616i)5-s + (−0.714 + 0.699i)6-s + (0.283 − 0.958i)7-s + (0.197 − 0.980i)8-s + (0.110 + 0.993i)9-s + (−0.562 − 0.826i)10-s + (0.428 + 0.903i)11-s + (−0.650 − 0.759i)12-s + (−0.862 + 0.506i)13-s + (0.937 + 0.346i)14-s + (−0.997 − 0.0663i)15-s + (0.964 + 0.262i)16-s + (−0.996 + 0.0883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.631 - 0.775i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.631 - 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3169562282 + 0.6666005092i\)
\(L(\frac12)\) \(\approx\) \(-0.3169562282 + 0.6666005092i\)
\(L(1)\) \(\approx\) \(0.5244600836 + 0.6837434884i\)
\(L(1)\) \(\approx\) \(0.5244600836 + 0.6837434884i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.0663 + 0.997i)T \)
3 \( 1 + (0.745 + 0.666i)T \)
5 \( 1 + (-0.787 + 0.616i)T \)
7 \( 1 + (0.283 - 0.958i)T \)
11 \( 1 + (0.428 + 0.903i)T \)
13 \( 1 + (-0.862 + 0.506i)T \)
17 \( 1 + (-0.996 + 0.0883i)T \)
19 \( 1 + (0.132 + 0.991i)T \)
23 \( 1 + (-0.850 - 0.525i)T \)
29 \( 1 + (-0.0883 + 0.996i)T \)
31 \( 1 + (-0.387 - 0.921i)T \)
37 \( 1 + (-0.958 - 0.283i)T \)
41 \( 1 + (-0.525 + 0.850i)T \)
43 \( 1 + (0.0663 + 0.997i)T \)
47 \( 1 + (0.0442 - 0.999i)T \)
53 \( 1 + (0.714 - 0.699i)T \)
59 \( 1 + (-0.0442 + 0.999i)T \)
61 \( 1 + (-0.408 - 0.912i)T \)
67 \( 1 + (-0.996 - 0.0883i)T \)
71 \( 1 + (0.197 + 0.980i)T \)
73 \( 1 + (-0.132 - 0.991i)T \)
79 \( 1 + (0.0221 + 0.999i)T \)
83 \( 1 + (0.544 - 0.839i)T \)
89 \( 1 + (0.387 - 0.921i)T \)
97 \( 1 + (0.970 + 0.240i)T \)
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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

−22.44750855650262217525083757165, −21.82076598711323017315348764650, −20.853562127393151318859507425074, −20.033909491377386594806208042001, −19.457736220375015807077939102711, −18.947376812435672559601673973, −17.90165248620125790036390727094, −17.26668455673815576832452969048, −15.7412243804814897957042984132, −15.05333304278164535904455624334, −13.93465171500559630529948191842, −13.2656176473218970188299013257, −12.17611755971125323020241820896, −11.956518124941342366678218887665, −10.92772579531301629821568511307, −9.44700692952442938559547188524, −8.80384161169322188576465717147, −8.26942756929899073070463610455, −7.24708804003572518912715083534, −5.70279964567844397923961697130, −4.63653498039999059313227185436, −3.55412715281050980637906693594, −2.661598832070889062395008858013, −1.696451050626747404360847141667, −0.34819872942507382201939672083, 1.949605319239547828881186038242, 3.58136280605381978398065275169, 4.229543115465113506114528182008, 4.87277502247100639882438393673, 6.56020724054364677852035201470, 7.337145456367063055333979083579, 7.92053871676104725054535706985, 8.929920930627021437154177168687, 9.958799385451112619947974874243, 10.51849129272250867679739130842, 11.83062598227167641131959351725, 13.06461901462165740060486022258, 14.16334871202103624012980566797, 14.60509754423230680247809364475, 15.18424059267059903035546274924, 16.2268318027044716721793754260, 16.77770762711194652385247534371, 17.84181519977393182309285260497, 18.7498851464253446635222166077, 19.839996726191411215286243718844, 20.09708142340675628817941996688, 21.48781275550126928958853972234, 22.4505864789977947398553910052, 22.83264678560297633616438502696, 23.97499834918767217624246808509

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