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Properties

Label 2-567-7.2-c1-0-25
Degree $2$
Conductor $567$
Sign $-0.979 + 0.202i$
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  + (1.29 − 2.24i)2-s + (−2.34 − 4.06i)4-s + (1.14 − 1.97i)5-s + (2.45 + 0.989i)7-s − 6.97·8-s + (−2.95 − 5.11i)10-s + (−1.47 − 2.56i)11-s + 4.26·13-s + (5.39 − 4.21i)14-s + (−4.32 + 7.49i)16-s + (−0.764 − 1.32i)17-s + (−3.69 + 6.39i)19-s − 10.7·20-s − 7.64·22-s + (−3.07 + 5.32i)23-s + ⋯
L(s)  = 1  + (0.914 − 1.58i)2-s + (−1.17 − 2.03i)4-s + (0.510 − 0.884i)5-s + (0.927 + 0.374i)7-s − 2.46·8-s + (−0.934 − 1.61i)10-s + (−0.445 − 0.771i)11-s + 1.18·13-s + (1.44 − 1.12i)14-s + (−1.08 + 1.87i)16-s + (−0.185 − 0.321i)17-s + (−0.846 + 1.46i)19-s − 2.39·20-s − 1.63·22-s + (−0.641 + 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.979 + 0.202i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.979 + 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.251325 - 2.45964i\)
\(L(\frac12)\) \(\approx\) \(0.251325 - 2.45964i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 - 0.989i)T \)
good2 \( 1 + (-1.29 + 2.24i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.14 + 1.97i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.47 + 2.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.26T + 13T^{2} \)
17 \( 1 + (0.764 + 1.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.69 - 6.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.07 - 5.32i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.34T + 29T^{2} \)
31 \( 1 + (3.11 + 5.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.58 - 6.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.89T + 41T^{2} \)
43 \( 1 - 0.834T + 43T^{2} \)
47 \( 1 + (-2.91 + 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.71 - 6.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.31 + 4.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.56 + 6.17i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 - 2.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.160T + 71T^{2} \)
73 \( 1 + (-0.190 - 0.329i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.97 + 6.88i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.29T + 83T^{2} \)
89 \( 1 + (3.02 - 5.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.32T + 97T^{2} \)
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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

−10.68986082259552018576136970486, −9.713508706636215716253762138513, −8.817472969513100316484949902159, −8.084518241731738478966449539545, −5.87412802022948566756514645091, −5.55843951532002473047011411575, −4.45285420372676828085228860088, −3.54442734790450551571233406248, −2.10520050949947398169414904179, −1.22478246562665312628652339678, 2.46050044921994584994125605386, 3.99735810439214277190157776092, 4.73890233202447965427104432941, 5.80235060419873928139586845029, 6.66331277950201310453804574105, 7.21519328289225071336763875979, 8.250978634363536774382127888785, 8.913872687796202343189649448678, 10.48182390528799860243115972118, 11.01599296895566861059864754773

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