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Properties

Label 1-569-569.30-r1-0-0
Degree $1$
Conductor $569$
Sign $0.937 - 0.347i$
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  + (−0.826 − 0.562i)2-s + (−0.878 + 0.477i)3-s + (0.367 + 0.930i)4-s + (−0.952 − 0.304i)5-s + (0.995 + 0.0993i)6-s + (−0.850 + 0.525i)7-s + (0.219 − 0.975i)8-s + (0.544 − 0.839i)9-s + (0.616 + 0.787i)10-s + (0.0552 + 0.998i)11-s + (−0.766 − 0.641i)12-s + (0.997 − 0.0663i)13-s + (0.999 + 0.0442i)14-s + (0.982 − 0.186i)15-s + (−0.730 + 0.683i)16-s + (−0.714 − 0.699i)17-s + ⋯
L(s)  = 1  + (−0.826 − 0.562i)2-s + (−0.878 + 0.477i)3-s + (0.367 + 0.930i)4-s + (−0.952 − 0.304i)5-s + (0.995 + 0.0993i)6-s + (−0.850 + 0.525i)7-s + (0.219 − 0.975i)8-s + (0.544 − 0.839i)9-s + (0.616 + 0.787i)10-s + (0.0552 + 0.998i)11-s + (−0.766 − 0.641i)12-s + (0.997 − 0.0663i)13-s + (0.999 + 0.0442i)14-s + (0.982 − 0.186i)15-s + (−0.730 + 0.683i)16-s + (−0.714 − 0.699i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.937 - 0.347i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ 0.937 - 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2536302277 - 0.04549265380i\)
\(L(\frac12)\) \(\approx\) \(0.2536302277 - 0.04549265380i\)
\(L(1)\) \(\approx\) \(0.3631952655 + 0.005167801231i\)
\(L(1)\) \(\approx\) \(0.3631952655 + 0.005167801231i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.826 - 0.562i)T \)
3 \( 1 + (-0.878 + 0.477i)T \)
5 \( 1 + (-0.952 - 0.304i)T \)
7 \( 1 + (-0.850 + 0.525i)T \)
11 \( 1 + (0.0552 + 0.998i)T \)
13 \( 1 + (0.997 - 0.0663i)T \)
17 \( 1 + (-0.714 - 0.699i)T \)
19 \( 1 + (-0.917 + 0.397i)T \)
23 \( 1 + (0.867 - 0.496i)T \)
29 \( 1 + (-0.999 - 0.0110i)T \)
31 \( 1 + (-0.336 + 0.941i)T \)
37 \( 1 + (-0.973 - 0.230i)T \)
41 \( 1 + (-0.964 - 0.262i)T \)
43 \( 1 + (-0.562 - 0.826i)T \)
47 \( 1 + (-0.925 + 0.377i)T \)
53 \( 1 + (0.0993 - 0.995i)T \)
59 \( 1 + (-0.377 - 0.925i)T \)
61 \( 1 + (-0.801 + 0.598i)T \)
67 \( 1 + (-0.714 + 0.699i)T \)
71 \( 1 + (-0.219 - 0.975i)T \)
73 \( 1 + (0.397 + 0.917i)T \)
79 \( 1 + (-0.980 + 0.197i)T \)
83 \( 1 + (0.315 + 0.948i)T \)
89 \( 1 + (-0.941 + 0.336i)T \)
97 \( 1 + (0.986 - 0.165i)T \)
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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

−23.27294726777420831752245400292, −22.70246765082039414655531912198, −21.605204560691852603328791241352, −20.19530844599929681388755949174, −19.31114794621237525454965982642, −18.94506519170119834275820878357, −18.17378080730161640361029394795, −16.95296827609963114710985973844, −16.64674644349731946958788178205, −15.70967267299107256570014181724, −15.09215399613322566748622321965, −13.6077792878784587382454730727, −12.99402779056492273992661572177, −11.563784994924709924241924101666, −10.98937178839899598909465894312, −10.4448332965549111883526446500, −9.02241834783816246217034280213, −8.17924625162688633990435965519, −7.2117288191748639675926549441, −6.49755768813380552264019669623, −5.89176350549643325104483356136, −4.47206597165650849350078975753, −3.27232547909718993921849198785, −1.557761687782416215695577400951, −0.37864733863508845128125792575, 0.26815937493923865369200487077, 1.69192707272272227919161874583, 3.24667427342183085888628672680, 4.0212160872861297643544463997, 5.07683320687668466911638932888, 6.57196380100115923260984327468, 7.115857585735666945393443952206, 8.58512423509706843874820595400, 9.13388562620415222545474623355, 10.17709453279612369535119128888, 10.96642774577189932518220912941, 11.741364592167187579691657817571, 12.52572083802605338784107323903, 13.03014102789553126444860890425, 15.124747624187452037290481006312, 15.66719034262839460010044640125, 16.37982505412989330982262835566, 17.07274736062055512482144987651, 18.140525851524485626523147989795, 18.72642828312215465347146130522, 19.642579603419815976780523603021, 20.55021803728610705659809891220, 21.08902555720437141789159913875, 22.353012165887246857490025237608, 22.735446238749720584350878003520

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