Properties

Label 2-567-189.184-c1-0-21
Degree $2$
Conductor $567$
Sign $-0.898 - 0.439i$
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  + (−0.306 − 1.73i)2-s + (−1.04 + 0.380i)4-s + (0.723 − 4.10i)5-s + (−1.55 − 2.14i)7-s + (−0.783 − 1.35i)8-s − 7.34·10-s + (0.428 + 2.43i)11-s + (0.108 + 0.0907i)13-s + (−3.24 + 3.35i)14-s + (−3.82 + 3.20i)16-s − 0.702·17-s + 6.46·19-s + (0.804 + 4.56i)20-s + (4.09 − 1.49i)22-s + (4.27 + 3.58i)23-s + ⋯
L(s)  = 1  + (−0.216 − 1.22i)2-s + (−0.522 + 0.190i)4-s + (0.323 − 1.83i)5-s + (−0.587 − 0.809i)7-s + (−0.277 − 0.479i)8-s − 2.32·10-s + (0.129 + 0.733i)11-s + (0.0299 + 0.0251i)13-s + (−0.866 + 0.897i)14-s + (−0.955 + 0.801i)16-s − 0.170·17-s + 1.48·19-s + (0.179 + 1.01i)20-s + (0.872 − 0.317i)22-s + (0.891 + 0.748i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.898 - 0.439i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.898 - 0.439i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.274300 + 1.18499i\)
\(L(\frac12)\) \(\approx\) \(0.274300 + 1.18499i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.55 + 2.14i)T \)
good2 \( 1 + (0.306 + 1.73i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-0.723 + 4.10i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.428 - 2.43i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-0.108 - 0.0907i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + 0.702T + 17T^{2} \)
19 \( 1 - 6.46T + 19T^{2} \)
23 \( 1 + (-4.27 - 3.58i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.100 - 0.0846i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-4.35 + 1.58i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (0.387 + 0.671i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.52 - 2.11i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (5.66 + 2.06i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (5.59 + 2.03i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (1.85 + 3.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.78 - 2.33i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.205 - 0.0746i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.66 + 15.1i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-7.04 + 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.36 - 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.00 - 5.68i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.84 + 1.54i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 0.904T + 89T^{2} \)
97 \( 1 + (1.73 + 0.629i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.951574256031742922953600266306, −9.655630965748964715662267767360, −8.943021322386112283220392870879, −7.76035372498485732141028090471, −6.61978920961302679129039197646, −5.26994947167051019953251747353, −4.37343278323517623600527690275, −3.30366706006272383559054303990, −1.69705089077670317189406148857, −0.76615755624957627370002816832, 2.62471816752154860446170147058, 3.25069629912819373209730808487, 5.28597508496755933162929206023, 6.15340071457632679466833866581, 6.66559129943111351264747194302, 7.41147739604558416451400849960, 8.435450286779019061007161998487, 9.373362793568486918814474963428, 10.22313755022302281898519386193, 11.27152665654107140421676913324

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