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Properties

Label 2-567-189.142-c1-0-20
Degree $2$
Conductor $567$
Sign $-0.609 - 0.792i$
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  + (0.324 − 1.83i)2-s + (−1.39 − 0.507i)4-s + (−1.37 − 0.500i)5-s + (−2.42 − 1.06i)7-s + (0.481 − 0.833i)8-s + (−1.36 + 2.36i)10-s + (−4.29 + 1.56i)11-s + (0.925 + 0.336i)13-s + (−2.73 + 4.11i)14-s + (−3.65 − 3.06i)16-s + (−0.620 + 1.07i)17-s + (−0.718 − 1.24i)19-s + (1.66 + 1.39i)20-s + (1.47 + 8.39i)22-s + (−0.165 − 0.939i)23-s + ⋯
L(s)  = 1  + (0.229 − 1.29i)2-s + (−0.697 − 0.253i)4-s + (−0.614 − 0.223i)5-s + (−0.916 − 0.400i)7-s + (0.170 − 0.294i)8-s + (−0.431 + 0.747i)10-s + (−1.29 + 0.470i)11-s + (0.256 + 0.0934i)13-s + (−0.731 + 1.09i)14-s + (−0.912 − 0.765i)16-s + (−0.150 + 0.260i)17-s + (−0.164 − 0.285i)19-s + (0.371 + 0.312i)20-s + (0.315 + 1.78i)22-s + (−0.0345 − 0.195i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.609 - 0.792i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.609 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.261705 + 0.531121i\)
\(L(\frac12)\) \(\approx\) \(0.261705 + 0.531121i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.42 + 1.06i)T \)
good2 \( 1 + (-0.324 + 1.83i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (1.37 + 0.500i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (4.29 - 1.56i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.925 - 0.336i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.620 - 1.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.718 + 1.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.165 + 0.939i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (5.18 - 1.88i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-6.28 - 2.28i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + 2.88T + 37T^{2} \)
41 \( 1 + (11.1 + 4.06i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.559 + 3.17i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-10.6 + 3.87i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (2.32 + 4.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.83 + 4.05i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (4.93 - 1.79i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.28 + 12.9i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-5.80 - 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.93T + 73T^{2} \)
79 \( 1 + (-2.45 + 13.9i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-7.82 + 2.84i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (0.307 + 0.532i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.991 - 5.62i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.40687723635092126871690502215, −9.699128511013842811863343964405, −8.583655362098257033224806171850, −7.50020029865341721660718942611, −6.63576913123284652755092985988, −5.14463857558515198669925286624, −4.08841753679281808571646102988, −3.27282823451466885490591443791, −2.12582876437662409550209399294, −0.29099629241638202707106027928, 2.61705175952304767664691150215, 3.84730933898383255923394810894, 5.18620149686664628898312526699, 5.92212045556001871942333948930, 6.77079737301435682334045582421, 7.72229252190792890182489503591, 8.257376393100829692729375884923, 9.336338707776266461570616528037, 10.43120052542690434435855011635, 11.27886732455342165475163873554

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