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Properties

Label 2-567-9.2-c2-0-41
Degree $2$
Conductor $567$
Sign $0.342 + 0.939i$
Analytic cond. $15.4496$
Root an. cond. $3.93060$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.13 + 0.653i)2-s + (−1.14 − 1.98i)4-s + (6.39 − 3.68i)5-s + (1.32 − 2.29i)7-s − 8.22i·8-s + 9.64·10-s + (2.26 + 1.30i)11-s + (3.17 + 5.50i)13-s + (2.99 − 1.72i)14-s + (0.791 − 1.37i)16-s − 12.1i·17-s − 10.2·19-s + (−14.6 − 8.45i)20-s + (1.70 + 2.95i)22-s + (3.72 − 2.15i)23-s + ⋯
L(s)  = 1  + (0.565 + 0.326i)2-s + (−0.286 − 0.496i)4-s + (1.27 − 0.737i)5-s + (0.188 − 0.327i)7-s − 1.02i·8-s + 0.964·10-s + (0.205 + 0.118i)11-s + (0.244 + 0.423i)13-s + (0.213 − 0.123i)14-s + (0.0494 − 0.0856i)16-s − 0.714i·17-s − 0.538·19-s + (−0.732 − 0.422i)20-s + (0.0776 + 0.134i)22-s + (0.161 − 0.0935i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(15.4496\)
Root analytic conductor: \(3.93060\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.692559780\)
\(L(\frac12)\) \(\approx\) \(2.692559780\)
\(L(2)\) 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 \( 1 + (-1.32 + 2.29i)T \)
good2 \( 1 + (-1.13 - 0.653i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-6.39 + 3.68i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.26 - 1.30i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.17 - 5.50i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 12.1iT - 289T^{2} \)
19 \( 1 + 10.2T + 361T^{2} \)
23 \( 1 + (-3.72 + 2.15i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (15.0 + 8.68i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (19.6 + 34.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 41.0T + 1.36e3T^{2} \)
41 \( 1 + (26.2 - 15.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-27.9 + 48.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-34.6 - 19.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 105. iT - 2.80e3T^{2} \)
59 \( 1 + (-35.8 + 20.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-10.2 + 17.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-13.5 - 23.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 67.8iT - 5.04e3T^{2} \)
73 \( 1 - 60.7T + 5.32e3T^{2} \)
79 \( 1 + (-31.6 + 54.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (77.8 + 44.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 63.1iT - 7.92e3T^{2} \)
97 \( 1 + (9.58 - 16.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.20334325027557098202646040735, −9.392505778526019738813391633964, −8.969611959784776775060212771997, −7.50569509083993858838137315534, −6.38215553100041157098726833184, −5.72927754948963897341401895550, −4.85546474040916440025510746626, −4.01496372730903795828391902665, −2.12314103722552351601455140838, −0.891471602368125060820477865977, 1.86365324411007821554550511374, 2.84837336238774977118073775440, 3.88477691245939242940907910901, 5.21355343690364496987892390171, 5.93511034339816659673445940777, 6.94769569516106100585314514506, 8.192267288043727576435949150371, 8.959744253315771670512090461137, 9.947873703196748954478112470644, 10.82438143745779929085844536330

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