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Properties

Label 2-567-63.16-c1-0-25
Degree $2$
Conductor $567$
Sign $-0.914 + 0.403i$
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  + (−0.768 − 1.33i)2-s + (−0.180 + 0.312i)4-s + 3.15·5-s + (−0.00900 − 2.64i)7-s − 2.51·8-s + (−2.42 − 4.20i)10-s − 5.74·11-s + (0.180 + 0.312i)13-s + (−3.51 + 2.04i)14-s + (2.29 + 3.97i)16-s + (−1.38 − 2.40i)17-s + (3.61 − 6.26i)19-s + (−0.569 + 0.986i)20-s + (4.41 + 7.65i)22-s − 0.824·23-s + ⋯
L(s)  = 1  + (−0.543 − 0.940i)2-s + (−0.0902 + 0.156i)4-s + 1.41·5-s + (−0.00340 − 0.999i)7-s − 0.890·8-s + (−0.767 − 1.32i)10-s − 1.73·11-s + (0.0500 + 0.0866i)13-s + (−0.939 + 0.546i)14-s + (0.573 + 0.994i)16-s + (−0.336 − 0.583i)17-s + (0.829 − 1.43i)19-s + (−0.127 + 0.220i)20-s + (0.941 + 1.63i)22-s − 0.171·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.914 + 0.403i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.914 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233358 - 1.10611i\)
\(L(\frac12)\) \(\approx\) \(0.233358 - 1.10611i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.00900 + 2.64i)T \)
good2 \( 1 + (0.768 + 1.33i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 3.15T + 5T^{2} \)
11 \( 1 + 5.74T + 11T^{2} \)
13 \( 1 + (-0.180 - 0.312i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.38 + 2.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.61 + 6.26i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.824T + 23T^{2} \)
29 \( 1 + (-2.13 + 3.70i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.49 + 4.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.74 - 6.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.66 - 2.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.93 + 6.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.74 - 3.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.45 - 2.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.19 - 2.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.60 + 2.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.949 - 1.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.60T + 71T^{2} \)
73 \( 1 + (-7.70 - 13.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.73 + 4.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.51 + 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.13 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.00 - 13.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.29992318424268155579719197327, −9.790660971972959791491264024783, −9.031539035672999137371760733817, −7.80758253405816207597327599441, −6.73531570276607856497753075266, −5.69976554306852441466993054319, −4.76513362730109879026010073757, −2.95283613175327988529467332635, −2.22640729744982323693711035457, −0.73265502211347225215863832170, 2.06487337656194591491898263297, 3.08238945896801249981852350841, 5.30538726006822154803095991814, 5.67541740765798151981750494603, 6.51171697947013990608226764909, 7.71131638999059715795011422194, 8.397186328081297908920516146243, 9.241786456417501081955196916796, 10.01728196318368895324188444798, 10.78510574422079054817129750191

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