↑

Properties

Label 2-567-189.101-c1-0-3
Degree $2$
Conductor $567$
Sign $-0.941 - 0.337i$
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  + (−0.572 + 0.682i)2-s + (0.209 + 1.18i)4-s + (1.06 − 0.891i)5-s + (−1.58 + 2.11i)7-s + (−2.47 − 1.42i)8-s + 1.23i·10-s + (0.371 − 0.443i)11-s + (−1.85 + 5.11i)13-s + (−0.539 − 2.29i)14-s + (0.126 − 0.0459i)16-s − 2.31·17-s − 3.42i·19-s + (1.28 + 1.07i)20-s + (0.0895 + 0.507i)22-s + (−2.28 + 6.28i)23-s + ⋯
L(s)  = 1  + (−0.405 + 0.482i)2-s + (0.104 + 0.593i)4-s + (0.474 − 0.398i)5-s + (−0.598 + 0.800i)7-s + (−0.874 − 0.505i)8-s + 0.390i·10-s + (0.112 − 0.133i)11-s + (−0.515 + 1.41i)13-s + (−0.144 − 0.613i)14-s + (0.0315 − 0.0114i)16-s − 0.560·17-s − 0.785i·19-s + (0.286 + 0.240i)20-s + (0.0190 + 0.108i)22-s + (−0.477 + 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.941 - 0.337i$
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: \((2,\ 567,\ (\ :1/2),\ -0.941 - 0.337i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.135458 + 0.779679i\)
\(L(\frac12)\) \(\approx\) \(0.135458 + 0.779679i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 2.11i)T \)
good2 \( 1 + (0.572 - 0.682i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-1.06 + 0.891i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-0.371 + 0.443i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.85 - 5.11i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 2.31T + 17T^{2} \)
19 \( 1 + 3.42iT - 19T^{2} \)
23 \( 1 + (2.28 - 6.28i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.224 - 0.616i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-1.71 + 0.302i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.542 + 0.939i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.37 + 0.500i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.681 - 3.86i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.07 - 6.06i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (8.16 + 4.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.76 + 1.73i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-13.3 - 2.34i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (6.31 - 5.29i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (13.6 - 7.86i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.07 + 1.19i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.05 + 1.72i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-11.2 + 4.10i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (-18.3 - 3.23i)T + (91.1 + 33.1i)T^{2} \)
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

−11.44053040166260086213690318138, −9.767676693317402213388810367836, −9.234317770530530168949259152581, −8.715552217992905422613094873812, −7.52330881854158552013715761614, −6.68268578217509180992969215033, −5.90564431698958446909573230591, −4.65307412796786683808458448610, −3.32772030350694637154155576983, −2.08097428104646676737145725987, 0.48182983884653254774562459056, 2.15187824145894616835849044277, 3.22729642694186410786049259773, 4.70203328076154678284141724882, 5.96303120463879737483085114244, 6.55165970368341978010152276697, 7.73724897452758972715369395589, 8.784338036971350106126292447147, 9.984049191168777018222002858503, 10.19804220939704749008128896854

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