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Properties

Label 2-567-63.4-c1-0-11
Degree $2$
Conductor $567$
Sign $0.852 - 0.523i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.298 − 0.517i)2-s + (0.821 + 1.42i)4-s + 2.09·5-s + (−1.51 + 2.17i)7-s + 2.17·8-s + (0.625 − 1.08i)10-s + 1.65·11-s + (0.213 − 0.368i)13-s + (0.672 + 1.43i)14-s + (−0.993 + 1.72i)16-s + (−3.03 + 5.26i)17-s + (−2.70 − 4.68i)19-s + (1.72 + 2.98i)20-s + (0.493 − 0.854i)22-s + 7.63·23-s + ⋯
L(s)  = 1  + (0.211 − 0.365i)2-s + (0.410 + 0.711i)4-s + 0.937·5-s + (−0.570 + 0.821i)7-s + 0.769·8-s + (0.197 − 0.342i)10-s + 0.497·11-s + (0.0590 − 0.102i)13-s + (0.179 + 0.382i)14-s + (−0.248 + 0.430i)16-s + (−0.736 + 1.27i)17-s + (−0.620 − 1.07i)19-s + (0.384 + 0.666i)20-s + (0.105 − 0.182i)22-s + 1.59·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.852 - 0.523i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.852 - 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95485 + 0.552319i\)
\(L(\frac12)\) \(\approx\) \(1.95485 + 0.552319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (1.51 - 2.17i)T \)
good2 \( 1 + (-0.298 + 0.517i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.09T + 5T^{2} \)
11 \( 1 - 1.65T + 11T^{2} \)
13 \( 1 + (-0.213 + 0.368i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.03 - 5.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.70 + 4.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.63T + 23T^{2} \)
29 \( 1 + (1.82 + 3.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.65 - 4.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.33 - 4.05i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.742 + 1.28i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.24 + 7.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.66 + 9.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.74 + 4.75i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.779 + 1.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.52 - 4.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.61 - 4.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 + (0.793 - 1.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.81 + 6.60i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.62 - 4.55i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.27 + 16.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.87 + 11.9i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.90346016427724123205589874889, −10.04602435531846445834160408521, −8.950839096868528240302654023851, −8.482400971490206234977124175784, −6.92979027423936939389668153284, −6.42026311374833768792947588113, −5.27199828502763864126910867973, −3.97948853875906604561371614183, −2.79824520569245795073051890212, −1.91739540594862740447902208929, 1.20521604460384562054738625045, 2.59445264740258704972782238107, 4.18061151449697946263027797306, 5.24495083173903142583352797011, 6.27173289633982927191082643735, 6.76179486319109368682123712219, 7.72307177264421460563199186896, 9.359906056182922003505838702008, 9.609133678267351238144318487905, 10.72780188620493318017049475940

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