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Properties

Label 2-567-63.59-c1-0-11
Degree $2$
Conductor $567$
Sign $-0.110 - 0.993i$
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  + (1.93 + 1.11i)2-s + (1.5 + 2.59i)4-s + (−0.5 + 2.59i)7-s + 2.23i·8-s + 4.47i·11-s + (1.5 + 0.866i)13-s + (−3.87 + 4.47i)14-s + (0.499 − 0.866i)16-s + (3.87 − 6.70i)17-s + (−3 + 1.73i)19-s + (−5.00 + 8.66i)22-s + 4.47i·23-s − 5·25-s + (1.93 + 3.35i)26-s + (−7.50 + 2.59i)28-s + (3.87 − 2.23i)29-s + ⋯
L(s)  = 1  + (1.36 + 0.790i)2-s + (0.750 + 1.29i)4-s + (−0.188 + 0.981i)7-s + 0.790i·8-s + 1.34i·11-s + (0.416 + 0.240i)13-s + (−1.03 + 1.19i)14-s + (0.124 − 0.216i)16-s + (0.939 − 1.62i)17-s + (−0.688 + 0.397i)19-s + (−1.06 + 1.84i)22-s + 0.932i·23-s − 25-s + (0.379 + 0.657i)26-s + (−1.41 + 0.490i)28-s + (0.719 − 0.415i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.110 - 0.993i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (458, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.110 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96844 + 2.20038i\)
\(L(\frac12)\) \(\approx\) \(1.96844 + 2.20038i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good2 \( 1 + (-1.93 - 1.11i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.87 + 6.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.47iT - 23T^{2} \)
29 \( 1 + (-3.87 + 2.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.87 + 6.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.87 + 6.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.87 + 2.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.87 + 6.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.5 + 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.94iT - 71T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.87 + 6.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.74 - 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.43655271021834977523210130036, −9.910260028151349991134174946707, −9.297956769334342171455768918488, −7.908242311504386547974617244888, −7.19436110783332259477414120107, −6.21144439715957212112110791888, −5.43430500912185319282891340992, −4.61289914684633452889720694876, −3.52091386849219783900137135396, −2.27372631983631384863324810657, 1.25713131595285212076105899498, 2.92836040090298976684038372553, 3.75634895091802207344630030279, 4.52964736841595479585401021137, 5.89120832122382174274368580677, 6.32292614949811262924127221532, 7.87564911597521824807741565801, 8.684520865285232893204800180668, 10.32180230069188330188012508444, 10.59916637241631611309125560271

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