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Properties

Label 1-569-569.59-r1-0-0
Degree $1$
Conductor $569$
Sign $0.663 + 0.748i$
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  + (−0.980 − 0.197i)2-s + (0.986 − 0.165i)3-s + (0.921 + 0.387i)4-s + (0.408 + 0.912i)5-s + (−0.999 − 0.0331i)6-s + (0.650 + 0.759i)7-s + (−0.826 − 0.562i)8-s + (0.945 − 0.325i)9-s + (−0.219 − 0.975i)10-s + (0.856 − 0.515i)11-s + (0.973 + 0.230i)12-s + (0.999 − 0.0221i)13-s + (−0.487 − 0.873i)14-s + (0.553 + 0.833i)15-s + (0.699 + 0.714i)16-s + (0.262 + 0.964i)17-s + ⋯
L(s)  = 1  + (−0.980 − 0.197i)2-s + (0.986 − 0.165i)3-s + (0.921 + 0.387i)4-s + (0.408 + 0.912i)5-s + (−0.999 − 0.0331i)6-s + (0.650 + 0.759i)7-s + (−0.826 − 0.562i)8-s + (0.945 − 0.325i)9-s + (−0.219 − 0.975i)10-s + (0.856 − 0.515i)11-s + (0.973 + 0.230i)12-s + (0.999 − 0.0221i)13-s + (−0.487 − 0.873i)14-s + (0.553 + 0.833i)15-s + (0.699 + 0.714i)16-s + (0.262 + 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.663 + 0.748i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ 0.663 + 0.748i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.561767250 + 1.152754058i\)
\(L(\frac12)\) \(\approx\) \(2.561767250 + 1.152754058i\)
\(L(1)\) \(\approx\) \(1.343896882 + 0.2316414740i\)
\(L(1)\) \(\approx\) \(1.343896882 + 0.2316414740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.980 - 0.197i)T \)
3 \( 1 + (0.986 - 0.165i)T \)
5 \( 1 + (0.408 + 0.912i)T \)
7 \( 1 + (0.650 + 0.759i)T \)
11 \( 1 + (0.856 - 0.515i)T \)
13 \( 1 + (0.999 - 0.0221i)T \)
17 \( 1 + (0.262 + 0.964i)T \)
19 \( 1 + (-0.377 + 0.925i)T \)
23 \( 1 + (0.641 + 0.766i)T \)
29 \( 1 + (-0.496 - 0.867i)T \)
31 \( 1 + (0.917 - 0.397i)T \)
37 \( 1 + (0.997 + 0.0773i)T \)
41 \( 1 + (-0.996 - 0.0883i)T \)
43 \( 1 + (0.197 + 0.980i)T \)
47 \( 1 + (-0.607 - 0.794i)T \)
53 \( 1 + (0.0331 - 0.999i)T \)
59 \( 1 + (-0.794 + 0.607i)T \)
61 \( 1 + (0.304 - 0.952i)T \)
67 \( 1 + (0.262 - 0.964i)T \)
71 \( 1 + (0.826 - 0.562i)T \)
73 \( 1 + (-0.925 - 0.377i)T \)
79 \( 1 + (-0.997 + 0.0663i)T \)
83 \( 1 + (0.807 - 0.589i)T \)
89 \( 1 + (-0.397 + 0.917i)T \)
97 \( 1 + (-0.998 + 0.0552i)T \)
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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

−23.28458823656410567584202268540, −21.708483431312021311322571116825, −20.74355447939603784831905055529, −20.42186171646769341642506253126, −19.815017955559404317422463904005, −18.760545661081887856104041908589, −17.884852150073312700321849541469, −17.05953246443128449106105800590, −16.35220097813254689103902069360, −15.4798485605673668988955837382, −14.50935671997799424072718878044, −13.80201795291555057759890203148, −12.814873037474452054269021286989, −11.5988484525968438194652416035, −10.62089343605197941009511713896, −9.69635902559799881661320808036, −8.92160989043723649960356693038, −8.430869147063056025903745310095, −7.36275299424907665685366141738, −6.59075013829988326821470448793, −5.0575345164856595912220303397, −4.16590941935872227670099456317, −2.74137775233982890472800400732, −1.51967402215583219102478390224, −0.93580412146493818728499396891, 1.357674351698378370668166661801, 1.9665062447421505486967476971, 3.10755222525153962913942045642, 3.840609455524198163628462178059, 5.97630850914070487171340080972, 6.524695897946850040602431667267, 7.85124238107116728239429720662, 8.33835857589708183667323483031, 9.258172257153655069518337464073, 10.02788638245481649769857175200, 11.068265635009385719297356620966, 11.75611965611085736632392158552, 12.96970145117007957431747934029, 13.97632408315566541829282195401, 14.98785155760948469478818151443, 15.2578601207094665880854025717, 16.6106108444092856187963444003, 17.57079225775196626119977535908, 18.401951690311546236660798213841, 18.956012717253564563893538379981, 19.48625317442657564270674121480, 20.72921090153128261039735421303, 21.29318089258987887055490594735, 21.85616640624623333492191384220, 23.26835159345546843050301917718

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