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Properties

Label 2-567-9.4-c1-0-4
Degree $2$
Conductor $567$
Sign $-0.766 + 0.642i$
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  + (1.32 + 2.29i)2-s + (−2.5 + 4.33i)4-s + (−1.32 + 2.29i)5-s + (0.5 + 0.866i)7-s − 7.93·8-s − 7·10-s + (−1.32 − 2.29i)11-s + (1 − 1.73i)13-s + (−1.32 + 2.29i)14-s + (−5.49 − 9.52i)16-s + 7·19-s + (−6.61 − 11.4i)20-s + (3.5 − 6.06i)22-s + (−3.96 + 6.87i)23-s + (−1 − 1.73i)25-s + 5.29·26-s + ⋯
L(s)  = 1  + (0.935 + 1.62i)2-s + (−1.25 + 2.16i)4-s + (−0.591 + 1.02i)5-s + (0.188 + 0.327i)7-s − 2.80·8-s − 2.21·10-s + (−0.398 − 0.690i)11-s + (0.277 − 0.480i)13-s + (−0.353 + 0.612i)14-s + (−1.37 − 2.38i)16-s + 1.60·19-s + (−1.47 − 2.56i)20-s + (0.746 − 1.29i)22-s + (−0.827 + 1.43i)23-s + (−0.200 − 0.346i)25-s + 1.03·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.588846 - 1.61784i\)
\(L(\frac12)\) \(\approx\) \(0.588846 - 1.61784i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-1.32 - 2.29i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.32 - 2.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.32 + 2.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (3.96 - 6.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.64 - 4.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 + (-1.32 + 2.29i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.93T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.93 - 13.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 18.5T + 89T^{2} \)
97 \( 1 + (-6 - 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.57280022164127027903671938191, −10.48402266287018830601639962228, −9.121424568335619773050275991253, −8.141219437061475374548472393855, −7.50931943475277156915929184087, −6.82795187870317546873145463931, −5.70955801295667906090318073610, −5.19369002616297607518684244554, −3.65264254732891797011775210284, −3.15808075737164071060245866665, 0.75660365681407308590198330003, 2.06866229084575812617631073718, 3.43460000464808613195212431072, 4.51728902817707379647895443485, 4.84283726508459786850027281540, 6.12140467144691533590486421017, 7.69679198748802826983559658472, 8.750637251906073843473889630461, 9.723719739852451926545900772075, 10.33212026710405397423568082552

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