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Properties

Label 2-567-21.20-c3-0-14
Degree $2$
Conductor $567$
Sign $-0.966 + 0.258i$
Analytic cond. $33.4540$
Root an. cond. $5.78395$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23i·2-s + 2.99·4-s − 5.51·5-s + (4.78 + 17.8i)7-s + 24.5i·8-s − 12.3i·10-s − 34.9i·11-s + 26.7i·13-s + (−40.0 + 10.6i)14-s − 31.0·16-s − 35.9·17-s + 78.0i·19-s − 16.5·20-s + 78.1·22-s + 57.8i·23-s + ⋯
L(s)  = 1  + 0.790i·2-s + 0.374·4-s − 0.493·5-s + (0.258 + 0.966i)7-s + 1.08i·8-s − 0.390i·10-s − 0.957i·11-s + 0.570i·13-s + (−0.764 + 0.204i)14-s − 0.485·16-s − 0.512·17-s + 0.942i·19-s − 0.184·20-s + 0.757·22-s + 0.524i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.966 + 0.258i$
Analytic conductor: \(33.4540\)
Root analytic conductor: \(5.78395\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :3/2),\ -0.966 + 0.258i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.250660217\)
\(L(\frac12)\) \(\approx\) \(1.250660217\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.78 - 17.8i)T \)
good2 \( 1 - 2.23iT - 8T^{2} \)
5 \( 1 + 5.51T + 125T^{2} \)
11 \( 1 + 34.9iT - 1.33e3T^{2} \)
13 \( 1 - 26.7iT - 2.19e3T^{2} \)
17 \( 1 + 35.9T + 4.91e3T^{2} \)
19 \( 1 - 78.0iT - 6.85e3T^{2} \)
23 \( 1 - 57.8iT - 1.21e4T^{2} \)
29 \( 1 + 221. iT - 2.43e4T^{2} \)
31 \( 1 - 145. iT - 2.97e4T^{2} \)
37 \( 1 - 290.T + 5.06e4T^{2} \)
41 \( 1 + 108.T + 6.89e4T^{2} \)
43 \( 1 + 348.T + 7.95e4T^{2} \)
47 \( 1 - 251.T + 1.03e5T^{2} \)
53 \( 1 + 160. iT - 1.48e5T^{2} \)
59 \( 1 + 734.T + 2.05e5T^{2} \)
61 \( 1 - 671. iT - 2.26e5T^{2} \)
67 \( 1 + 688.T + 3.00e5T^{2} \)
71 \( 1 - 577. iT - 3.57e5T^{2} \)
73 \( 1 + 37.4iT - 3.89e5T^{2} \)
79 \( 1 - 855.T + 4.93e5T^{2} \)
83 \( 1 + 1.23e3T + 5.71e5T^{2} \)
89 \( 1 + 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + 1.15e3iT - 9.12e5T^{2} \)
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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

−11.10425287641582730358408067902, −9.815447492145070413846163930609, −8.657607916802960060584161646457, −8.171299166852178093382774046270, −7.27027119662254839709467259412, −6.12712774446732483439406486005, −5.68600481480396815693474241052, −4.37650154435363944065491430417, −2.99856695377092030778291329791, −1.78398998373607764263129688243, 0.34627417694242575993826724379, 1.64362494997300852509525774311, 2.87404290187202028053282205992, 3.98817664176383833527833306263, 4.80259179738718761043640313180, 6.42801700581326057162128009165, 7.24116514785343840685343471340, 7.902316559768356931171583421820, 9.257675266897885920747395620319, 10.13109123934694878989110134836

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