Properties

Label 2-567-189.185-c1-0-16
Degree $2$
Conductor $567$
Sign $0.957 + 0.288i$
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  + (2.26 − 0.399i)2-s + (3.10 − 1.12i)4-s + (1.73 − 0.630i)5-s + (0.678 + 2.55i)7-s + (2.59 − 1.49i)8-s + (3.67 − 2.12i)10-s + (0.0420 − 0.115i)11-s + (−0.0159 − 0.0437i)13-s + (2.56 + 5.52i)14-s + (0.221 − 0.185i)16-s + (−0.0786 − 0.136i)17-s + (−6.57 − 3.79i)19-s + (4.65 − 3.90i)20-s + (0.0490 − 0.278i)22-s + (0.957 + 0.168i)23-s + ⋯
L(s)  = 1  + (1.60 − 0.282i)2-s + (1.55 − 0.564i)4-s + (0.774 − 0.281i)5-s + (0.256 + 0.966i)7-s + (0.916 − 0.528i)8-s + (1.16 − 0.670i)10-s + (0.0126 − 0.0347i)11-s + (−0.00441 − 0.0121i)13-s + (0.684 + 1.47i)14-s + (0.0553 − 0.0464i)16-s + (−0.0190 − 0.0330i)17-s + (−1.50 − 0.871i)19-s + (1.04 − 0.874i)20-s + (0.0104 − 0.0593i)22-s + (0.199 + 0.0352i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.957 + 0.288i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.75223 - 0.552789i\)
\(L(\frac12)\) \(\approx\) \(3.75223 - 0.552789i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.678 - 2.55i)T \)
good2 \( 1 + (-2.26 + 0.399i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (-1.73 + 0.630i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (-0.0420 + 0.115i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.0159 + 0.0437i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.0786 + 0.136i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.57 + 3.79i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.957 - 0.168i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.33 + 9.16i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (1.56 + 4.29i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 - 9.37T + 37T^{2} \)
41 \( 1 + (5.76 - 2.09i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.67 - 9.52i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (3.91 + 1.42i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.141 - 0.0818i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.06 + 7.60i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.03 + 2.85i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.70 - 9.68i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.863 - 0.498i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.18iT - 73T^{2} \)
79 \( 1 + (-0.451 - 2.56i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-6.02 - 2.19i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-5.30 + 9.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.2 + 2.33i)T + (91.1 - 33.1i)T^{2} \)
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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

−11.23867132407840807918055453020, −9.931096885602281262514844868116, −9.055238508952963410500168864152, −8.020776659954959731552513483100, −6.40671237753668445267022131579, −6.00994780218864509526785945890, −4.99781324965828755249579522270, −4.26686122778447241850859393195, −2.77105758171864804872233117999, −2.01884909826845663714853655316, 1.92079203703569662850673492684, 3.27993189174166056655332526822, 4.25275416768493133362230988494, 5.11383548629553427667496624602, 6.18787878111850721086329341009, 6.74302514416816568743676217267, 7.75397295374767048621911616418, 9.008036886809033024994934008020, 10.37344334303716012255190472495, 10.73340849880166134464746189422

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