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Properties

Label 2-567-189.101-c1-0-18
Degree $2$
Conductor $567$
Sign $0.883 + 0.467i$
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  + (−0.204 + 0.244i)2-s + (0.329 + 1.86i)4-s + (2.18 − 1.83i)5-s + (0.468 − 2.60i)7-s + (−1.07 − 0.621i)8-s + 0.911i·10-s + (2.19 − 2.61i)11-s + (1.17 − 3.21i)13-s + (0.539 + 0.647i)14-s + (−3.19 + 1.16i)16-s − 4.66·17-s − 5.52i·19-s + (4.15 + 3.48i)20-s + (0.188 + 1.06i)22-s + (−0.470 + 1.29i)23-s + ⋯
L(s)  = 1  + (−0.144 + 0.172i)2-s + (0.164 + 0.934i)4-s + (0.979 − 0.821i)5-s + (0.176 − 0.984i)7-s + (−0.380 − 0.219i)8-s + 0.288i·10-s + (0.660 − 0.787i)11-s + (0.324 − 0.891i)13-s + (0.144 + 0.173i)14-s + (−0.798 + 0.290i)16-s − 1.13·17-s − 1.26i·19-s + (0.929 + 0.779i)20-s + (0.0402 + 0.228i)22-s + (−0.0980 + 0.269i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.883 + 0.467i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.883 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58988 - 0.394717i\)
\(L(\frac12)\) \(\approx\) \(1.58988 - 0.394717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.468 + 2.60i)T \)
good2 \( 1 + (0.204 - 0.244i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-2.18 + 1.83i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-2.19 + 2.61i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.17 + 3.21i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 4.66T + 17T^{2} \)
19 \( 1 + 5.52iT - 19T^{2} \)
23 \( 1 + (0.470 - 1.29i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-2.81 - 7.73i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.17 + 0.384i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.882 + 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.96 - 1.80i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.57 - 8.95i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.532 - 3.01i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-1.30 - 0.756i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.44 - 2.34i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-9.69 - 1.70i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.34 - 6.16i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-9.64 + 5.56i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.11 - 2.95i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.75 - 7.34i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.72 + 2.44i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + (-2.47 - 0.436i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.82320518279723144635415477488, −9.568332093710154473715464661327, −8.842828245595547983048973329528, −8.183821048459539009802462954393, −7.04361012043917757150878301096, −6.28857196979184007997089661525, −5.03660564819721512387875953503, −4.00787608474881048705594774809, −2.78028129317611599279539498106, −1.05788820968901586333144173400, 1.86278688305777841367503971781, 2.36553266539104967867817689360, 4.24306962219828359506527671606, 5.50647942415082334827120104087, 6.34280069170542049016288044940, 6.76512397153131128993347794594, 8.419790142422286335641953165168, 9.367811568125326782248067502846, 9.871618201501547334313758453823, 10.68994923120096023598380256689

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