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Properties

Label 1-569-569.58-r1-0-0
Degree $1$
Conductor $569$
Sign $-0.249 - 0.968i$
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  + (0.0883 + 0.996i)2-s + (−0.186 − 0.982i)3-s + (−0.984 + 0.176i)4-s + (0.633 + 0.773i)5-s + (0.962 − 0.273i)6-s + (0.616 − 0.787i)7-s + (−0.262 − 0.964i)8-s + (−0.930 + 0.367i)9-s + (−0.714 + 0.699i)10-s + (0.752 − 0.658i)11-s + (0.356 + 0.934i)12-s + (−0.650 + 0.759i)13-s + (0.839 + 0.544i)14-s + (0.641 − 0.766i)15-s + (0.937 − 0.346i)16-s + (−0.801 − 0.598i)17-s + ⋯
L(s)  = 1  + (0.0883 + 0.996i)2-s + (−0.186 − 0.982i)3-s + (−0.984 + 0.176i)4-s + (0.633 + 0.773i)5-s + (0.962 − 0.273i)6-s + (0.616 − 0.787i)7-s + (−0.262 − 0.964i)8-s + (−0.930 + 0.367i)9-s + (−0.714 + 0.699i)10-s + (0.752 − 0.658i)11-s + (0.356 + 0.934i)12-s + (−0.650 + 0.759i)13-s + (0.839 + 0.544i)14-s + (0.641 − 0.766i)15-s + (0.937 − 0.346i)16-s + (−0.801 − 0.598i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.249 - 0.968i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ -0.249 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5785146887 - 0.7467740296i\)
\(L(\frac12)\) \(\approx\) \(0.5785146887 - 0.7467740296i\)
\(L(1)\) \(\approx\) \(0.9451030694 + 0.07729121906i\)
\(L(1)\) \(\approx\) \(0.9451030694 + 0.07729121906i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.0883 + 0.996i)T \)
3 \( 1 + (-0.186 - 0.982i)T \)
5 \( 1 + (0.633 + 0.773i)T \)
7 \( 1 + (0.616 - 0.787i)T \)
11 \( 1 + (0.752 - 0.658i)T \)
13 \( 1 + (-0.650 + 0.759i)T \)
17 \( 1 + (-0.801 - 0.598i)T \)
19 \( 1 + (0.820 - 0.571i)T \)
23 \( 1 + (-0.457 - 0.888i)T \)
29 \( 1 + (-0.143 + 0.989i)T \)
31 \( 1 + (-0.251 - 0.967i)T \)
37 \( 1 + (-0.121 - 0.992i)T \)
41 \( 1 + (0.952 + 0.304i)T \)
43 \( 1 + (-0.996 - 0.0883i)T \)
47 \( 1 + (-0.948 + 0.315i)T \)
53 \( 1 + (0.273 + 0.962i)T \)
59 \( 1 + (0.315 + 0.948i)T \)
61 \( 1 + (-0.467 - 0.883i)T \)
67 \( 1 + (-0.801 + 0.598i)T \)
71 \( 1 + (0.262 - 0.964i)T \)
73 \( 1 + (0.571 + 0.820i)T \)
79 \( 1 + (-0.850 - 0.525i)T \)
83 \( 1 + (-0.515 + 0.856i)T \)
89 \( 1 + (-0.967 - 0.251i)T \)
97 \( 1 + (-0.833 + 0.553i)T \)
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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

−22.833768045020635682371873466895, −22.17548641018607081607588739577, −21.57262742454099205556965620220, −20.8940864596035331218173363974, −20.12355373739557015350883349696, −19.57396002749374388542086249371, −17.95827664509635809324373899713, −17.64143873211241273327455814365, −16.83727495830292912598411050578, −15.54586114946376422714254336163, −14.77982288586265466144249165536, −13.976437538344558876493470363036, −12.84539605193291978147803025653, −11.99353929735230440870042660671, −11.451200040314255670693800265235, −10.161140923259771583839044415806, −9.697380179450539462863603381925, −8.88065566775783348734541260590, −8.09978596943593999118136016069, −6.02846498283854384120376196029, −5.21364375984112371921675062807, −4.6328093532122613074622258306, −3.56746447176399182318186735684, −2.30471341969945272461642796479, −1.37741058132587585281325734337, 0.23204204251557268207692597669, 1.48982833437849618712050798764, 2.81583946913654034846484515064, 4.214245069803465426941312334755, 5.31463599477500516027871335239, 6.32548569611748080933279548433, 6.98941501631910042397118105887, 7.52422524574015995439384278748, 8.72442105061462216206826929425, 9.578293692601544825685417680452, 10.92048923943124646921145040752, 11.66090023382016518964094473932, 12.91842884647752028701380705090, 13.93427495385919461028209252593, 14.05244575343985492191984577681, 14.89090601052577089438849092129, 16.43071483528734726506405548630, 16.9267786997106178433986106010, 17.91119425653198175763389388038, 18.18831014790816620025365667466, 19.22935081538760912074694159705, 20.10402766032425137890347745501, 21.54297604630707951265457039844, 22.28661487056135735162713547202, 22.855517696205319693726786513105

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