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Properties

Label 2-567-21.20-c3-0-1
Degree $2$
Conductor $567$
Sign $0.613 - 0.789i$
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  − 4.53i·2-s − 12.5·4-s + 0.137·5-s + (14.6 + 11.3i)7-s + 20.6i·8-s − 0.623i·10-s − 10.1i·11-s − 15.1i·13-s + (51.5 − 66.2i)14-s − 6.90·16-s − 88.2·17-s + 66.1i·19-s − 1.72·20-s − 45.9·22-s − 57.4i·23-s + ⋯
L(s)  = 1  − 1.60i·2-s − 1.56·4-s + 0.0122·5-s + (0.789 + 0.613i)7-s + 0.911i·8-s − 0.0197i·10-s − 0.277i·11-s − 0.323i·13-s + (0.983 − 1.26i)14-s − 0.107·16-s − 1.25·17-s + 0.798i·19-s − 0.0192·20-s − 0.445·22-s − 0.520i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.613 - 0.789i$
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: \((2,\ 567,\ (\ :3/2),\ 0.613 - 0.789i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2315481163\)
\(L(\frac12)\) \(\approx\) \(0.2315481163\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14.6 - 11.3i)T \)
good2 \( 1 + 4.53iT - 8T^{2} \)
5 \( 1 - 0.137T + 125T^{2} \)
11 \( 1 + 10.1iT - 1.33e3T^{2} \)
13 \( 1 + 15.1iT - 2.19e3T^{2} \)
17 \( 1 + 88.2T + 4.91e3T^{2} \)
19 \( 1 - 66.1iT - 6.85e3T^{2} \)
23 \( 1 + 57.4iT - 1.21e4T^{2} \)
29 \( 1 + 118. iT - 2.43e4T^{2} \)
31 \( 1 - 307. iT - 2.97e4T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + 441.T + 6.89e4T^{2} \)
43 \( 1 + 93.4T + 7.95e4T^{2} \)
47 \( 1 - 311.T + 1.03e5T^{2} \)
53 \( 1 - 321. iT - 1.48e5T^{2} \)
59 \( 1 - 574.T + 2.05e5T^{2} \)
61 \( 1 + 732. iT - 2.26e5T^{2} \)
67 \( 1 + 63.2T + 3.00e5T^{2} \)
71 \( 1 - 621. iT - 3.57e5T^{2} \)
73 \( 1 - 95.2iT - 3.89e5T^{2} \)
79 \( 1 + 850.T + 4.93e5T^{2} \)
83 \( 1 + 140.T + 5.71e5T^{2} \)
89 \( 1 + 1.22e3T + 7.04e5T^{2} \)
97 \( 1 - 775. iT - 9.12e5T^{2} \)
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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

−10.59828906476977804584631180044, −9.913813573540512766535180934643, −8.768445144829949728593974584337, −8.354983158012507854881696512209, −6.85779897307172644128896066842, −5.52672781300028988218762230926, −4.55127567115498202199376738904, −3.51674273539459926501353437421, −2.35678262159362272399453206611, −1.50665023439554263417568590516, 0.06623178549254475330345909975, 1.97787782810586130762078023465, 4.00225886335307747586250404855, 4.80390314237707653608401808723, 5.68822081724164084960247932729, 6.88086087632028509409580169169, 7.25580150590872127425274660480, 8.300078315354662171562559257679, 8.928393188709340004393474385370, 9.970337648093883333351355315669

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