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Properties

Label 2-567-189.104-c1-0-18
Degree $2$
Conductor $567$
Sign $-0.631 + 0.775i$
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  + (0.422 − 1.16i)2-s + (0.362 + 0.304i)4-s + (0.253 − 1.44i)5-s + (−2.19 − 1.47i)7-s + (2.64 − 1.52i)8-s + (−1.56 − 0.903i)10-s + (−4.32 + 0.762i)11-s + (−1.72 − 4.72i)13-s + (−2.64 + 1.92i)14-s + (−0.491 − 2.78i)16-s + (0.691 − 1.19i)17-s + (4.13 − 2.38i)19-s + (0.530 − 0.445i)20-s + (−0.941 + 5.34i)22-s + (3.82 − 4.55i)23-s + ⋯
L(s)  = 1  + (0.298 − 0.820i)2-s + (0.181 + 0.152i)4-s + (0.113 − 0.644i)5-s + (−0.829 − 0.558i)7-s + (0.935 − 0.540i)8-s + (−0.494 − 0.285i)10-s + (−1.30 + 0.229i)11-s + (−0.477 − 1.31i)13-s + (−0.706 + 0.513i)14-s + (−0.122 − 0.696i)16-s + (0.167 − 0.290i)17-s + (0.948 − 0.547i)19-s + (0.118 − 0.0996i)20-s + (−0.200 + 1.13i)22-s + (0.797 − 0.950i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.631 + 0.775i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.687194 - 1.44629i\)
\(L(\frac12)\) \(\approx\) \(0.687194 - 1.44629i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 1.47i)T \)
good2 \( 1 + (-0.422 + 1.16i)T + (-1.53 - 1.28i)T^{2} \)
5 \( 1 + (-0.253 + 1.44i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (4.32 - 0.762i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (1.72 + 4.72i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.691 + 1.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.13 + 2.38i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.82 + 4.55i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.96 - 5.40i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.06 - 4.84i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (-4.81 + 8.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.26 + 0.459i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.613 + 3.47i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (5.55 - 4.66i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 0.811iT - 53T^{2} \)
59 \( 1 + (0.933 - 5.29i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-5.13 - 6.11i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (4.21 - 1.53i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-11.6 - 6.74i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.23 - 3.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.07 + 2.93i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-4.57 - 1.66i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-0.219 - 0.380i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.66 + 0.646i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.52151999030252527304400428903, −9.887675509815850711914986798032, −8.815099234010712576523717313536, −7.52858287822339934227481288344, −7.12469933747876862643242516393, −5.47986953120789909075811356520, −4.73084011209907687978745680685, −3.30666566178416937796409101196, −2.65814542647231133163960244605, −0.818244271767134767990751694302, 2.13256775181817699605715483833, 3.25104595647705902497003431428, 4.84357445771451259919720526440, 5.74443520058743051452599794843, 6.47669683264186545165757287057, 7.31281641148607974426370226509, 8.075204225130882541519474776859, 9.464554635268071929521508155141, 10.04985480619594503049453729790, 11.11679254696250888708976579763

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