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Properties

Label 2-567-63.47-c1-0-6
Degree $2$
Conductor $567$
Sign $-0.585 - 0.810i$
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  + (−1.58 + 0.917i)2-s + (0.682 − 1.18i)4-s + 3.80·5-s + (−2.43 + 1.03i)7-s − 1.16i·8-s + (−6.04 + 3.49i)10-s + 1.16i·11-s + (−4.79 + 2.77i)13-s + (2.91 − 3.88i)14-s + (2.43 + 4.21i)16-s + (1.58 + 2.75i)17-s + (−0.546 − 0.315i)19-s + (2.59 − 4.50i)20-s + (−1.06 − 1.85i)22-s + 1.76i·23-s + ⋯
L(s)  = 1  + (−1.12 + 0.648i)2-s + (0.341 − 0.590i)4-s + 1.70·5-s + (−0.919 + 0.392i)7-s − 0.412i·8-s + (−1.91 + 1.10i)10-s + 0.351i·11-s + (−1.33 + 0.768i)13-s + (0.778 − 1.03i)14-s + (0.608 + 1.05i)16-s + (0.385 + 0.667i)17-s + (−0.125 − 0.0724i)19-s + (0.580 − 1.00i)20-s + (−0.227 − 0.394i)22-s + 0.367i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.585 - 0.810i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.585 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360385 + 0.704851i\)
\(L(\frac12)\) \(\approx\) \(0.360385 + 0.704851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.43 - 1.03i)T \)
good2 \( 1 + (1.58 - 0.917i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 3.80T + 5T^{2} \)
11 \( 1 - 1.16iT - 11T^{2} \)
13 \( 1 + (4.79 - 2.77i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.58 - 2.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.546 + 0.315i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.76iT - 23T^{2} \)
29 \( 1 + (-4.18 - 2.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.70 - 2.13i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.11 - 3.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.28 - 3.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.61 - 6.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.27 + 2.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0627 + 0.0362i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.49 - 6.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.00 - 4.04i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.54 - 4.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.76iT - 71T^{2} \)
73 \( 1 + (-4.84 + 2.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.54 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.08 + 8.80i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.77 + 10.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.596 - 0.344i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.22630738310165129175119825214, −10.06300383411785824681742749411, −9.300995786890100745717134878674, −8.709179783634068231293321712054, −7.41687514857795689445976367804, −6.54941781632695240377795313382, −6.03215902022224197661981550463, −4.78061433086636029815566531691, −2.92248372835269572133708893660, −1.63102344620754218889415868199, 0.64759303471121258745501010350, 2.21559845984140918453685917343, 2.98306136267249859094617014626, 5.01870558546105164520557500413, 5.87675195438714719558167171912, 6.91170006312194026241277956652, 8.015213303863084818532661130109, 9.160745723141093607086438803823, 9.696183780988217957306274243049, 10.18847215890019711790428671416

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