Properties

Label 2-567-189.104-c1-0-0
Degree $2$
Conductor $567$
Sign $0.602 - 0.797i$
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  + (0.878 − 2.41i)2-s + (−3.52 − 2.95i)4-s + (−0.588 + 3.33i)5-s + (−2.64 + 0.108i)7-s + (−5.79 + 3.34i)8-s + (7.53 + 4.35i)10-s + (−4.55 + 0.803i)11-s + (0.619 + 1.70i)13-s + (−2.06 + 6.47i)14-s + (1.38 + 7.86i)16-s + (−1.40 + 2.43i)17-s + (−0.586 + 0.338i)19-s + (11.9 − 10.0i)20-s + (−2.06 + 11.7i)22-s + (2.29 − 2.74i)23-s + ⋯
L(s)  = 1  + (0.621 − 1.70i)2-s + (−1.76 − 1.47i)4-s + (−0.263 + 1.49i)5-s + (−0.999 + 0.0411i)7-s + (−2.04 + 1.18i)8-s + (2.38 + 1.37i)10-s + (−1.37 + 0.242i)11-s + (0.171 + 0.472i)13-s + (−0.550 + 1.73i)14-s + (0.346 + 1.96i)16-s + (−0.340 + 0.590i)17-s + (−0.134 + 0.0777i)19-s + (2.67 − 2.24i)20-s + (−0.439 + 2.49i)22-s + (0.479 − 0.571i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.602 - 0.797i$
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: \((2,\ 567,\ (\ :1/2),\ 0.602 - 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378014 + 0.188131i\)
\(L(\frac12)\) \(\approx\) \(0.378014 + 0.188131i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.108i)T \)
good2 \( 1 + (-0.878 + 2.41i)T + (-1.53 - 1.28i)T^{2} \)
5 \( 1 + (0.588 - 3.33i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (4.55 - 0.803i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-0.619 - 1.70i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.40 - 2.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.586 - 0.338i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.29 + 2.74i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.546 + 1.50i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (1.99 - 2.37i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (0.898 - 1.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (11.6 - 4.22i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.0534 + 0.303i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-2.22 + 1.86i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 4.31iT - 53T^{2} \)
59 \( 1 + (-1.34 + 7.61i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-0.783 - 0.933i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.826 + 0.300i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (11.6 + 6.72i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (11.2 - 6.52i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.7 - 5.01i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (6.55 + 2.38i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-8.53 - 14.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.81 + 1.02i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.78106043950801111972432766393, −10.37438354779987310282207774548, −9.730967953907713749981898815969, −8.512319279097672028342699929195, −7.09271170885065788168517093125, −6.19800912253034127604085368292, −4.94905930004281706614592034450, −3.71232005153650674388134169802, −3.01151932607369530675291168927, −2.18696894664431837488234570882, 0.18563888310864128349320131330, 3.21540400574854098346314090862, 4.41409291222324595416102656975, 5.27023778613900552034428180750, 5.80623175275881158912108886862, 7.04985280912272130294016486404, 7.78462441806198973906405751473, 8.665995348360740242499436793532, 9.182983186684719128411282359216, 10.39534356600611496314912907906

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