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Properties

Label 2-567-189.101-c1-0-13
Degree $2$
Conductor $567$
Sign $0.707 + 0.706i$
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  + (1.08 − 1.29i)2-s + (−0.151 − 0.861i)4-s + (−1.13 + 0.954i)5-s + (2.51 − 0.806i)7-s + (1.65 + 0.953i)8-s + 2.51i·10-s + (0.0836 − 0.0996i)11-s + (−0.311 + 0.855i)13-s + (1.69 − 4.15i)14-s + (4.68 − 1.70i)16-s + 5.63·17-s + 0.0959i·19-s + (0.995 + 0.835i)20-s + (−0.0383 − 0.217i)22-s + (2.22 − 6.10i)23-s + ⋯
L(s)  = 1  + (0.770 − 0.918i)2-s + (−0.0759 − 0.430i)4-s + (−0.508 + 0.426i)5-s + (0.952 − 0.304i)7-s + (0.584 + 0.337i)8-s + 0.796i·10-s + (0.0252 − 0.0300i)11-s + (−0.0863 + 0.237i)13-s + (0.453 − 1.10i)14-s + (1.17 − 0.426i)16-s + 1.36·17-s + 0.0220i·19-s + (0.222 + 0.186i)20-s + (−0.00816 − 0.0463i)22-s + (0.463 − 1.27i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.707 + 0.706i$
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: \((2,\ 567,\ (\ :1/2),\ 0.707 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17567 - 0.900193i\)
\(L(\frac12)\) \(\approx\) \(2.17567 - 0.900193i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.51 + 0.806i)T \)
good2 \( 1 + (-1.08 + 1.29i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.13 - 0.954i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-0.0836 + 0.0996i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.311 - 0.855i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 5.63T + 17T^{2} \)
19 \( 1 - 0.0959iT - 19T^{2} \)
23 \( 1 + (-2.22 + 6.10i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.238 + 0.656i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (8.96 - 1.58i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.72 + 2.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.04 - 0.744i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.37 + 7.82i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.18 - 12.3i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (1.26 + 0.731i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.96 + 2.17i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (7.69 + 1.35i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (11.0 - 9.25i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.38 - 1.37i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.94 - 4.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.310 - 0.260i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-4.54 + 1.65i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 + (-10.2 - 1.81i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.94794153591166280952569705315, −10.22219128153497934155535666262, −8.907478279410727901880435465494, −7.75585874329104675982099350392, −7.30535882705606566817256234549, −5.70099263276652285943408761937, −4.70323055663123055272626689140, −3.85013781954473142609416516835, −2.87165055074091272061744366922, −1.50948337347884916201169703283, 1.45337346347671756736537846465, 3.45166422945669078328998973093, 4.56593526585735610827209128561, 5.31083805863860981045267496317, 6.03876549645048499123169610536, 7.50727382389764127365056358729, 7.72562591334060748999700018443, 8.858998623326579646666131006130, 9.969983826879436142019919399408, 10.99186528243556800838339112198

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