Properties

Label 2-567-567.104-c1-0-41
Degree $2$
Conductor $567$
Sign $0.999 + 0.0318i$
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  + (1.82 + 1.36i)2-s + (−1.59 − 0.679i)3-s + (0.916 + 3.06i)4-s + (−1.51 − 0.999i)5-s + (−1.98 − 3.41i)6-s + (0.330 − 2.62i)7-s + (−0.933 + 2.56i)8-s + (2.07 + 2.16i)9-s + (−1.41 − 3.89i)10-s + (2.17 − 4.32i)11-s + (0.620 − 5.50i)12-s + (2.75 + 1.19i)13-s + (4.17 − 4.34i)14-s + (1.74 + 2.62i)15-s + (0.145 − 0.0954i)16-s + (4.41 + 3.70i)17-s + ⋯
L(s)  = 1  + (1.29 + 0.962i)2-s + (−0.919 − 0.392i)3-s + (0.458 + 1.53i)4-s + (−0.679 − 0.446i)5-s + (−0.811 − 1.39i)6-s + (0.125 − 0.992i)7-s + (−0.329 + 0.906i)8-s + (0.692 + 0.721i)9-s + (−0.448 − 1.23i)10-s + (0.655 − 1.30i)11-s + (0.179 − 1.58i)12-s + (0.765 + 0.330i)13-s + (1.11 − 1.16i)14-s + (0.449 + 0.677i)15-s + (0.0362 − 0.0238i)16-s + (1.07 + 0.898i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.999 + 0.0318i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.999 + 0.0318i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06132 - 0.0328462i\)
\(L(\frac12)\) \(\approx\) \(2.06132 - 0.0328462i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.59 + 0.679i)T \)
7 \( 1 + (-0.330 + 2.62i)T \)
good2 \( 1 + (-1.82 - 1.36i)T + (0.573 + 1.91i)T^{2} \)
5 \( 1 + (1.51 + 0.999i)T + (1.98 + 4.59i)T^{2} \)
11 \( 1 + (-2.17 + 4.32i)T + (-6.56 - 8.82i)T^{2} \)
13 \( 1 + (-2.75 - 1.19i)T + (8.92 + 9.45i)T^{2} \)
17 \( 1 + (-4.41 - 3.70i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (0.529 + 0.630i)T + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (4.23 + 3.99i)T + (1.33 + 22.9i)T^{2} \)
29 \( 1 + (0.770 - 6.59i)T + (-28.2 - 6.68i)T^{2} \)
31 \( 1 + (-1.42 + 6.01i)T + (-27.7 - 13.9i)T^{2} \)
37 \( 1 + (-0.124 + 0.703i)T + (-34.7 - 12.6i)T^{2} \)
41 \( 1 + (3.15 + 4.23i)T + (-11.7 + 39.2i)T^{2} \)
43 \( 1 + (-0.270 - 4.63i)T + (-42.7 + 4.99i)T^{2} \)
47 \( 1 + (3.76 - 0.891i)T + (42.0 - 21.0i)T^{2} \)
53 \( 1 + (3.19 + 1.84i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.56 - 1.28i)T + (35.2 - 47.3i)T^{2} \)
61 \( 1 + (-12.9 - 3.88i)T + (50.9 + 33.5i)T^{2} \)
67 \( 1 + (-10.6 + 1.24i)T + (65.1 - 15.4i)T^{2} \)
71 \( 1 + (-3.04 - 8.37i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.0272 - 0.0749i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (3.92 - 5.26i)T + (-22.6 - 75.6i)T^{2} \)
83 \( 1 + (-0.613 + 0.823i)T + (-23.8 - 79.5i)T^{2} \)
89 \( 1 + (9.05 + 3.29i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-6.69 - 10.1i)T + (-38.4 + 89.0i)T^{2} \)
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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

−11.12430791648046662621358804293, −10.14969666812071331030447816519, −8.360733311393945975431666144950, −7.85893435022359106170671409833, −6.72930317678648421523798692810, −6.18907904817900667783101110280, −5.29839020399182330024652369123, −4.13423446120905525235582709323, −3.73074283894241001143158483942, −0.997187789164026611395014599740, 1.68689054616131113367816557433, 3.23244661969464567408438513995, 4.03044580822489635697681688117, 5.02967263292553639293654466238, 5.74280189377355090974846822937, 6.73963526090666234004785685342, 7.971961123005891844346794930979, 9.558732652551345933201081324812, 10.12834585303584065779514465690, 11.25963791693552918042482326424

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