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Properties

Label 1-569-569.15-r1-0-0
Degree $1$
Conductor $569$
Sign $0.468 - 0.883i$
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  + (−0.467 − 0.883i)2-s + (−0.989 − 0.143i)3-s + (−0.562 + 0.826i)4-s + (0.154 − 0.988i)5-s + (0.336 + 0.941i)6-s + (0.873 + 0.487i)7-s + (0.993 + 0.110i)8-s + (0.958 + 0.283i)9-s + (−0.945 + 0.325i)10-s + (−0.356 + 0.934i)11-s + (0.675 − 0.737i)12-s + (0.683 − 0.730i)13-s + (0.0221 − 0.999i)14-s + (−0.294 + 0.955i)15-s + (−0.367 − 0.930i)16-s + (−0.387 − 0.921i)17-s + ⋯
L(s)  = 1  + (−0.467 − 0.883i)2-s + (−0.989 − 0.143i)3-s + (−0.562 + 0.826i)4-s + (0.154 − 0.988i)5-s + (0.336 + 0.941i)6-s + (0.873 + 0.487i)7-s + (0.993 + 0.110i)8-s + (0.958 + 0.283i)9-s + (−0.945 + 0.325i)10-s + (−0.356 + 0.934i)11-s + (0.675 − 0.737i)12-s + (0.683 − 0.730i)13-s + (0.0221 − 0.999i)14-s + (−0.294 + 0.955i)15-s + (−0.367 − 0.930i)16-s + (−0.387 − 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.468 - 0.883i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ 0.468 - 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.068372697 - 0.6425041556i\)
\(L(\frac12)\) \(\approx\) \(1.068372697 - 0.6425041556i\)
\(L(1)\) \(\approx\) \(0.6725602082 - 0.3362201164i\)
\(L(1)\) \(\approx\) \(0.6725602082 - 0.3362201164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.467 - 0.883i)T \)
3 \( 1 + (-0.989 - 0.143i)T \)
5 \( 1 + (0.154 - 0.988i)T \)
7 \( 1 + (0.873 + 0.487i)T \)
11 \( 1 + (-0.356 + 0.934i)T \)
13 \( 1 + (0.683 - 0.730i)T \)
17 \( 1 + (-0.387 - 0.921i)T \)
19 \( 1 + (0.982 + 0.186i)T \)
23 \( 1 + (-0.607 - 0.794i)T \)
29 \( 1 + (0.377 + 0.925i)T \)
31 \( 1 + (0.845 + 0.534i)T \)
37 \( 1 + (0.962 - 0.273i)T \)
41 \( 1 + (0.991 + 0.132i)T \)
43 \( 1 + (0.883 + 0.467i)T \)
47 \( 1 + (-0.833 + 0.553i)T \)
53 \( 1 + (-0.941 + 0.336i)T \)
59 \( 1 + (0.553 + 0.833i)T \)
61 \( 1 + (-0.894 - 0.448i)T \)
67 \( 1 + (-0.387 + 0.921i)T \)
71 \( 1 + (-0.993 + 0.110i)T \)
73 \( 1 + (-0.186 + 0.982i)T \)
79 \( 1 + (0.773 + 0.633i)T \)
83 \( 1 + (0.230 - 0.973i)T \)
89 \( 1 + (0.534 + 0.845i)T \)
97 \( 1 + (-0.457 - 0.888i)T \)
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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

−23.43395077758880387286914618244, −22.53264355266305346287271840100, −21.73998550020322531434870943604, −20.97706616361561149707230842373, −19.432724520359583104411266565292, −18.7159863870969781584683197468, −17.8753774636732757652212888421, −17.51403974344781980751788157620, −16.496497730470231590574695943399, −15.75909922277850047988699943253, −14.968043620130254425716505876097, −13.88043561904117165034135140636, −13.452520420113685157165753719477, −11.59509755677003293225777479395, −11.07695372926420598750587862745, −10.354780723127516715444682656551, −9.442658336276954185100021338736, −8.09240085503913017377703444563, −7.42263522477033270687633231097, −6.25834498092747848224135274233, −5.93824336747687075256107850330, −4.69031040044397716288111217782, −3.75828481597938742101207623457, −1.791677597707929328962298947761, −0.65678156225717291905795876809, 0.774840209012279651126196887028, 1.483613827020443057915903909066, 2.65501965840685114339734336790, 4.43842494232530107899699001109, 4.85718363177361660173395404889, 5.85802830483145554552260007915, 7.43535598497713147403580127474, 8.161901509493112888184802305983, 9.2037933837275130252390749603, 10.063422089132250575518152050320, 10.98684171623844348330528712556, 11.798952440101279087057601624422, 12.452981148636028737343523111913, 13.07330999577669460760858455732, 14.17246205615968786857632159029, 15.80958840755669883698060645524, 16.2440100356745707768914072744, 17.46305211401177943176562347022, 17.995635639341743675390211197167, 18.29088742363145682125773071109, 19.731205553480597178681026476975, 20.63916474613512862477362340895, 20.9590023066546232157891983703, 22.02587471382668398927980863940, 22.75837632178514805280319241515

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