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Properties

Label 1-569-569.7-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.212 - 0.977i$
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  + (−0.487 + 0.873i)2-s + (−0.773 + 0.633i)3-s + (−0.525 − 0.850i)4-s + (−0.367 − 0.930i)5-s + (−0.176 − 0.984i)6-s + (0.110 − 0.993i)7-s + (0.999 − 0.0442i)8-s + (0.197 − 0.980i)9-s + (0.991 + 0.132i)10-s + (0.714 − 0.699i)11-s + (0.945 + 0.325i)12-s + (0.598 − 0.801i)13-s + (0.814 + 0.580i)14-s + (0.873 + 0.487i)15-s + (−0.448 + 0.894i)16-s + (−0.154 − 0.988i)17-s + ⋯
L(s)  = 1  + (−0.487 + 0.873i)2-s + (−0.773 + 0.633i)3-s + (−0.525 − 0.850i)4-s + (−0.367 − 0.930i)5-s + (−0.176 − 0.984i)6-s + (0.110 − 0.993i)7-s + (0.999 − 0.0442i)8-s + (0.197 − 0.980i)9-s + (0.991 + 0.132i)10-s + (0.714 − 0.699i)11-s + (0.945 + 0.325i)12-s + (0.598 − 0.801i)13-s + (0.814 + 0.580i)14-s + (0.873 + 0.487i)15-s + (−0.448 + 0.894i)16-s + (−0.154 − 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.212 - 0.977i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.212 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2999648784 - 0.3720262292i\)
\(L(\frac12)\) \(\approx\) \(0.2999648784 - 0.3720262292i\)
\(L(1)\) \(\approx\) \(0.5614238332 + 0.01063766275i\)
\(L(1)\) \(\approx\) \(0.5614238332 + 0.01063766275i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.487 + 0.873i)T \)
3 \( 1 + (-0.773 + 0.633i)T \)
5 \( 1 + (-0.367 - 0.930i)T \)
7 \( 1 + (0.110 - 0.993i)T \)
11 \( 1 + (0.714 - 0.699i)T \)
13 \( 1 + (0.598 - 0.801i)T \)
17 \( 1 + (-0.154 - 0.988i)T \)
19 \( 1 + (-0.850 - 0.525i)T \)
23 \( 1 + (-0.544 + 0.839i)T \)
29 \( 1 + (-0.988 - 0.154i)T \)
31 \( 1 + (0.0883 - 0.996i)T \)
37 \( 1 + (0.993 + 0.110i)T \)
41 \( 1 + (-0.839 - 0.544i)T \)
43 \( 1 + (0.487 + 0.873i)T \)
47 \( 1 + (-0.650 - 0.759i)T \)
53 \( 1 + (0.176 + 0.984i)T \)
59 \( 1 + (0.650 + 0.759i)T \)
61 \( 1 + (-0.903 - 0.428i)T \)
67 \( 1 + (-0.154 + 0.988i)T \)
71 \( 1 + (0.999 + 0.0442i)T \)
73 \( 1 + (0.850 + 0.525i)T \)
79 \( 1 + (-0.937 - 0.346i)T \)
83 \( 1 + (-0.219 + 0.975i)T \)
89 \( 1 + (-0.0883 - 0.996i)T \)
97 \( 1 + (0.683 + 0.730i)T \)
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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

−23.23844663683413009263133174577, −22.500034572779788481769772100245, −21.90999621732274218386649985251, −21.18656964481978759246228843718, −19.83403919380249456929937727057, −19.13149821376055030154746471581, −18.52360250377091611593895052658, −17.98623234244505745692505987196, −17.06924777885512947697782249055, −16.21647979173003305160689800417, −14.96006535439321786406123000316, −14.09775382117809293310138534364, −12.80075237937273376818812673626, −12.24912699326607156342490237789, −11.467125957580382851782753270035, −10.85714151576785145025559595768, −9.93152437385021309522172903457, −8.72058885073085145972072508173, −7.937492009203414503762711309831, −6.76673682991790261765423971948, −6.13606900411948228341555999452, −4.59903620510116319225132777929, −3.653936022126272915352956323354, −2.199884759753099969157654649163, −1.68775629630557767379346424763, 0.37426481001038047629356425360, 1.198772873328439999699492899424, 3.7698101696861638069902267925, 4.40123829715821543868384894920, 5.40205091817834622602688694499, 6.17584119548436249007706976414, 7.26308299992305166609790020271, 8.21622536785424557550303557355, 9.164876660925249992700477405352, 9.873992254272766294934924045062, 10.99807063871274646585671435810, 11.53412059161497450232513662748, 13.03144542317965612220200159020, 13.69783132341088201244450800755, 14.93946801932437496753494095786, 15.70195654072955456902113415617, 16.4575040567866108448609504894, 16.97050268944557116273202188121, 17.60084080520780802655452987035, 18.610603996155887304224800224721, 19.84523838834411899008442188033, 20.32051902904704409728276126346, 21.41445540897988905557864036987, 22.55466757484501728295830425649, 23.14088553920425274756933494956

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