↑

Properties

Label 2-567-189.173-c1-0-9
Degree $2$
Conductor $567$
Sign $0.814 - 0.580i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.170 + 0.469i)2-s + (1.34 − 1.12i)4-s + (0.000430 − 0.000361i)5-s + (1.75 + 1.98i)7-s + (1.62 + 0.936i)8-s + (0.000243 + 0.000140i)10-s + (−3.04 + 3.62i)11-s + (1.77 + 2.10i)13-s + (−0.632 + 1.16i)14-s + (0.445 − 2.52i)16-s + (0.400 − 0.692i)17-s + (4.50 − 2.60i)19-s + (0.000170 − 0.000969i)20-s + (−2.22 − 0.808i)22-s + (−0.542 + 1.49i)23-s + ⋯
L(s)  = 1  + (0.120 + 0.331i)2-s + (0.670 − 0.562i)4-s + (0.000192 − 0.000161i)5-s + (0.661 + 0.749i)7-s + (0.573 + 0.331i)8-s + (7.68e−5 + 4.43e−5i)10-s + (−0.917 + 1.09i)11-s + (0.490 + 0.585i)13-s + (−0.168 + 0.310i)14-s + (0.111 − 0.631i)16-s + (0.0970 − 0.168i)17-s + (1.03 − 0.597i)19-s + (3.82e−5 − 0.000216i)20-s + (−0.473 − 0.172i)22-s + (−0.113 + 0.310i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.814 - 0.580i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.814 - 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86517 + 0.596768i\)
\(L(\frac12)\) \(\approx\) \(1.86517 + 0.596768i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.75 - 1.98i)T \)
good2 \( 1 + (-0.170 - 0.469i)T + (-1.53 + 1.28i)T^{2} \)
5 \( 1 + (-0.000430 + 0.000361i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (3.04 - 3.62i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.77 - 2.10i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.400 + 0.692i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.50 + 2.60i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.542 - 1.49i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-5.40 + 6.44i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (5.19 + 6.18i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 - 2.05T + 37T^{2} \)
41 \( 1 + (-5.20 + 4.36i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (6.39 - 2.32i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-5.68 - 4.76i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (9.11 - 5.26i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.29 + 13.0i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.33 - 2.78i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (1.99 + 0.727i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (7.56 - 4.37i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.18iT - 73T^{2} \)
79 \( 1 + (2.42 - 0.881i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (1.47 + 1.23i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-1.61 - 2.80i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.614 - 1.68i)T + (-74.3 + 62.3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.04253701506920961337800587797, −9.879202194021273089087080266186, −9.223200773610279776610262318439, −7.83049915237275231152393739840, −7.38624814832237875612092957055, −6.16657525831804170332154840662, −5.35713262002035124346732409098, −4.53992578088505567947714249800, −2.69265244327573992162232696399, −1.67939343024668676504444581968, 1.26881512390512282569415770544, 2.88037582682140506389573222925, 3.68463478050456295196223641422, 4.99494114696580989655516798692, 6.09518841350340478875670851176, 7.24078789607781765527414900998, 8.004595977451416507159399330224, 8.595796842661874541211278994958, 10.33705662637309346469305409825, 10.59526528659115376587757817264

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