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Properties

Label 2-567-7.2-c1-0-24
Degree $2$
Conductor $567$
Sign $-0.605 + 0.795i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.500 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−2.5 + 0.866i)7-s − 3·8-s + (0.499 + 0.866i)10-s + (−2.5 − 4.33i)11-s − 5·13-s + (0.500 − 2.59i)14-s + (0.500 − 0.866i)16-s + (−1.5 − 2.59i)17-s + (−0.5 + 0.866i)19-s + 20-s + 5·22-s + (−1.5 + 2.59i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.250 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.944 + 0.327i)7-s − 1.06·8-s + (0.158 + 0.273i)10-s + (−0.753 − 1.30i)11-s − 1.38·13-s + (0.133 − 0.694i)14-s + (0.125 − 0.216i)16-s + (−0.363 − 0.630i)17-s + (−0.114 + 0.198i)19-s + 0.223·20-s + 1.06·22-s + (−0.312 + 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (-6.5 + 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9T + 97T^{2} \)
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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

−10.21656143679400210428457110927, −9.326091098605186756926143498073, −8.684016413149816931467332311475, −7.72344960370949364598645747823, −6.92206390354702262883459612684, −5.94680635297489605012493830811, −5.13306347590381444544408360813, −3.40928784563991957555486050511, −2.56913021709831103433907185803, 0, 2.09722158097719399578055374252, 2.83500348986569341134381138791, 4.39878002081084735989505116194, 5.59544006671016637352202132287, 6.72164552928728548187375234778, 7.23952305521182142772308156990, 8.666529902372777410911250752005, 9.833952699299365717202941903333, 10.07413883922760963736345076231

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