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Properties

Label 2-567-21.20-c3-0-38
Degree $2$
Conductor $567$
Sign $-0.264 - 0.964i$
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  + 1.27i·2-s + 6.37·4-s + 3.19·5-s + (17.8 − 4.90i)7-s + 18.3i·8-s + 4.07i·10-s + 43.9i·11-s + 75.7i·13-s + (6.26 + 22.7i)14-s + 27.5·16-s − 104.·17-s − 74.7i·19-s + 20.3·20-s − 56.1·22-s + 53.8i·23-s + ⋯
L(s)  = 1  + 0.451i·2-s + 0.796·4-s + 0.285·5-s + (0.964 − 0.264i)7-s + 0.810i·8-s + 0.128i·10-s + 1.20i·11-s + 1.61i·13-s + (0.119 + 0.435i)14-s + 0.430·16-s − 1.49·17-s − 0.902i·19-s + 0.227·20-s − 0.544·22-s + 0.488i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.264 - 0.964i$
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: \((2,\ 567,\ (\ :3/2),\ -0.264 - 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.613202145\)
\(L(\frac12)\) \(\approx\) \(2.613202145\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.8 + 4.90i)T \)
good2 \( 1 - 1.27iT - 8T^{2} \)
5 \( 1 - 3.19T + 125T^{2} \)
11 \( 1 - 43.9iT - 1.33e3T^{2} \)
13 \( 1 - 75.7iT - 2.19e3T^{2} \)
17 \( 1 + 104.T + 4.91e3T^{2} \)
19 \( 1 + 74.7iT - 6.85e3T^{2} \)
23 \( 1 - 53.8iT - 1.21e4T^{2} \)
29 \( 1 - 30.4iT - 2.43e4T^{2} \)
31 \( 1 - 128. iT - 2.97e4T^{2} \)
37 \( 1 - 46.1T + 5.06e4T^{2} \)
41 \( 1 - 426.T + 6.89e4T^{2} \)
43 \( 1 - 332.T + 7.95e4T^{2} \)
47 \( 1 + 340.T + 1.03e5T^{2} \)
53 \( 1 + 235. iT - 1.48e5T^{2} \)
59 \( 1 + 544.T + 2.05e5T^{2} \)
61 \( 1 + 371. iT - 2.26e5T^{2} \)
67 \( 1 - 106.T + 3.00e5T^{2} \)
71 \( 1 - 974. iT - 3.57e5T^{2} \)
73 \( 1 + 576. iT - 3.89e5T^{2} \)
79 \( 1 - 21.3T + 4.93e5T^{2} \)
83 \( 1 + 337.T + 5.71e5T^{2} \)
89 \( 1 - 1.02e3T + 7.04e5T^{2} \)
97 \( 1 - 73.3iT - 9.12e5T^{2} \)
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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

−10.85919104303564754345863215118, −9.594478819415410929642260947079, −8.829214870448010617320443590580, −7.65137003007305725295270026010, −7.01981883867407587283521097085, −6.29093038241068861035402692271, −4.95999404130602359108053888720, −4.25272145597826339279081411064, −2.30952715331856354553642596206, −1.70143733054997059309216124422, 0.72607890852488527266900809974, 2.04884475404659122311439286800, 2.97559271903419510183983825488, 4.23959531075201432603268473835, 5.72042707823843521552760123534, 6.14227565448999282109409020602, 7.63189231086117332128592448891, 8.174747424657080943112709056108, 9.271088561141278129205762442671, 10.45282108547238341834366875442

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