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Properties

Label 2-567-189.20-c1-0-0
Degree $2$
Conductor $567$
Sign $0.797 + 0.603i$
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  + (−0.924 − 2.53i)2-s + (−4.05 + 3.40i)4-s + (−0.268 − 1.52i)5-s + (−1.64 + 2.07i)7-s + (7.72 + 4.45i)8-s + (−3.61 + 2.08i)10-s + (−3.67 − 0.647i)11-s + (−0.202 + 0.555i)13-s + (6.78 + 2.26i)14-s + (2.34 − 13.2i)16-s + (2.21 + 3.83i)17-s + (5.88 + 3.39i)19-s + (6.26 + 5.26i)20-s + (1.74 + 9.92i)22-s + (−1.22 − 1.45i)23-s + ⋯
L(s)  = 1  + (−0.653 − 1.79i)2-s + (−2.02 + 1.70i)4-s + (−0.119 − 0.680i)5-s + (−0.621 + 0.783i)7-s + (2.72 + 1.57i)8-s + (−1.14 + 0.659i)10-s + (−1.10 − 0.195i)11-s + (−0.0560 + 0.153i)13-s + (1.81 + 0.604i)14-s + (0.585 − 3.32i)16-s + (0.537 + 0.930i)17-s + (1.34 + 0.778i)19-s + (1.40 + 1.17i)20-s + (0.373 + 2.11i)22-s + (−0.255 − 0.304i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.797 + 0.603i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.797 + 0.603i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.574991 - 0.192895i\)
\(L(\frac12)\) \(\approx\) \(0.574991 - 0.192895i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 2.07i)T \)
good2 \( 1 + (0.924 + 2.53i)T + (-1.53 + 1.28i)T^{2} \)
5 \( 1 + (0.268 + 1.52i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (3.67 + 0.647i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.202 - 0.555i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.21 - 3.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.88 - 3.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.22 + 1.45i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-0.530 - 1.45i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.97 - 4.74i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (1.87 + 3.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.26 + 0.459i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.06 - 6.03i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-4.70 - 3.95i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 2.96iT - 53T^{2} \)
59 \( 1 + (-0.751 - 4.26i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (6.03 - 7.19i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-4.30 - 1.56i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (4.85 - 2.80i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.50 + 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.68 - 2.79i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (4.97 - 1.80i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (2.95 - 5.11i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-18.3 - 3.24i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.44141689291711450848975781161, −10.04064607015395102787918009829, −9.036889348964432597706398913913, −8.462041013534802202173333257668, −7.63428479141151602927470617204, −5.73719401164871095155025429522, −4.72487195890837077356877387459, −3.45554912456602215778878881384, −2.62928589468556197994899651717, −1.21864960282029052713054281755, 0.50654743403140890516738069820, 3.17260316832941447693566770790, 4.71926867317760084747054402444, 5.54416667649046894980686267344, 6.62024233234897226545027607652, 7.38441526465663229757091035186, 7.69688534659606111510231111253, 8.930329467348561191894716410169, 9.962056731106068404861722848846, 10.18508936491047767613185203168

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