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Properties

Label 2-567-189.101-c1-0-7
Degree $2$
Conductor $567$
Sign $0.388 - 0.921i$
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  + (0.575 − 0.686i)2-s + (0.207 + 1.17i)4-s + (−0.100 + 0.0844i)5-s + (−1.70 + 2.02i)7-s + (2.48 + 1.43i)8-s + 0.117i·10-s + (−2.57 + 3.06i)11-s + (1.02 − 2.81i)13-s + (0.409 + 2.33i)14-s + (0.160 − 0.0583i)16-s + 0.344·17-s + 4.89i·19-s + (−0.120 − 0.101i)20-s + (0.623 + 3.53i)22-s + (−2.18 + 5.98i)23-s + ⋯
L(s)  = 1  + (0.407 − 0.485i)2-s + (0.103 + 0.589i)4-s + (−0.0450 + 0.0377i)5-s + (−0.643 + 0.765i)7-s + (0.877 + 0.506i)8-s + 0.0372i·10-s + (−0.776 + 0.925i)11-s + (0.284 − 0.781i)13-s + (0.109 + 0.623i)14-s + (0.0401 − 0.0145i)16-s + 0.0835·17-s + 1.12i·19-s + (−0.0269 − 0.0226i)20-s + (0.132 + 0.753i)22-s + (−0.454 + 1.24i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.388 - 0.921i$
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: \((2,\ 567,\ (\ :1/2),\ 0.388 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29482 + 0.859162i\)
\(L(\frac12)\) \(\approx\) \(1.29482 + 0.859162i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 2.02i)T \)
good2 \( 1 + (-0.575 + 0.686i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (0.100 - 0.0844i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (2.57 - 3.06i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.02 + 2.81i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 0.344T + 17T^{2} \)
19 \( 1 - 4.89iT - 19T^{2} \)
23 \( 1 + (2.18 - 5.98i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (2.29 + 6.31i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-8.59 + 1.51i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-3.64 + 6.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.04 - 3.29i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.350 + 1.98i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.771 - 4.37i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (5.49 + 3.17i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.167 + 0.0608i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (4.60 + 0.811i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-8.83 + 7.40i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.373 - 0.215i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.31 + 0.761i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.81 - 1.52i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.81 + 2.48i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + (12.2 + 2.15i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.08588713418175977327489475816, −10.04544308240615120374999319415, −9.369861816202430010696958259648, −7.87866029598669253140034148046, −7.74232577694249051222610613927, −6.18328708688724152406635578689, −5.30430231199346743143882923959, −4.04879542221110573447827054510, −3.07625517438671302293502878482, −2.07601474934392622228448724545, 0.77940904299551353733557515373, 2.70420600552102013397232466580, 4.14071372593315284187325394092, 4.96542820298439860673468112779, 6.24805095306810248900648264816, 6.63176169052752317791108186244, 7.74125335800292790888891697547, 8.790936902410863926170932251280, 9.858813202062799156446336634165, 10.59847500035629628737911774684

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