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Properties

Label 2-567-63.47-c1-0-18
Degree $2$
Conductor $567$
Sign $0.996 + 0.0845i$
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  + (0.568 − 0.328i)2-s + (−0.784 + 1.35i)4-s + 3.31·5-s + (0.799 − 2.52i)7-s + 2.34i·8-s + (1.88 − 1.08i)10-s − 2.34i·11-s + (1.36 − 0.790i)13-s + (−0.373 − 1.69i)14-s + (−0.799 − 1.38i)16-s + (−0.568 − 0.984i)17-s + (3.85 + 2.22i)19-s + (−2.59 + 4.5i)20-s + (−0.769 − 1.33i)22-s + 9.39i·23-s + ⋯
L(s)  = 1  + (0.402 − 0.232i)2-s + (−0.392 + 0.679i)4-s + 1.48·5-s + (0.302 − 0.953i)7-s + 0.828i·8-s + (0.595 − 0.343i)10-s − 0.706i·11-s + (0.379 − 0.219i)13-s + (−0.0997 − 0.453i)14-s + (−0.199 − 0.346i)16-s + (−0.137 − 0.238i)17-s + (0.884 + 0.510i)19-s + (−0.580 + 1.00i)20-s + (−0.164 − 0.284i)22-s + 1.95i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.996 + 0.0845i$
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: \((2,\ 567,\ (\ :1/2),\ 0.996 + 0.0845i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15554 - 0.0913027i\)
\(L(\frac12)\) \(\approx\) \(2.15554 - 0.0913027i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.799 + 2.52i)T \)
good2 \( 1 + (-0.568 + 0.328i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 3.31T + 5T^{2} \)
11 \( 1 + 2.34iT - 11T^{2} \)
13 \( 1 + (-1.36 + 0.790i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.568 + 0.984i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.85 - 2.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 9.39iT - 23T^{2} \)
29 \( 1 + (3.16 + 1.82i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.33 - 3.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.58 + 4.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.82 + 8.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.08 + 1.87i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.79 - 4.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.70 - 5.02i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.08 - 1.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.46 - 2.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.38 + 9.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.39iT - 71T^{2} \)
73 \( 1 + (9.25 - 5.34i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.616 + 1.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.518 + 0.898i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.73 - 6.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.7 + 6.77i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.80341460896495596946350738460, −9.813982838114130083101399734219, −9.132648235938647453361563002902, −8.051997986013505527536034342546, −7.21634991818448790769151720247, −5.86986663696388130050779693939, −5.22883373091706473252086475376, −3.94324839383084792346351179326, −2.98281906370579302341694783776, −1.47553767699880752436394069949, 1.52404821146312053683932146360, 2.65311825737885376181299607092, 4.55386471150907069836113280300, 5.21120243634865245524639321382, 6.16009841632918199179903292469, 6.63970487515648459726082161432, 8.300860898122324701486364248414, 9.245092150514986085389826520741, 9.760080828375009571267232731704, 10.52889657371547905345919578256

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