↑

Properties

Label 2-567-189.101-c1-0-8
Degree $2$
Conductor $567$
Sign $-0.0318 - 0.999i$
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  + (0.159 − 0.190i)2-s + (0.336 + 1.90i)4-s + (−1.62 + 1.36i)5-s + (2.64 − 0.158i)7-s + (0.848 + 0.489i)8-s + 0.529i·10-s + (0.919 − 1.09i)11-s + (−1.30 + 3.59i)13-s + (0.392 − 0.528i)14-s + (−3.41 + 1.24i)16-s − 0.218·17-s − 2.96i·19-s + (−3.15 − 2.64i)20-s + (−0.0618 − 0.350i)22-s + (−2.64 + 7.26i)23-s + ⋯
L(s)  = 1  + (0.113 − 0.134i)2-s + (0.168 + 0.954i)4-s + (−0.728 + 0.611i)5-s + (0.998 − 0.0599i)7-s + (0.300 + 0.173i)8-s + 0.167i·10-s + (0.277 − 0.330i)11-s + (−0.362 + 0.996i)13-s + (0.104 − 0.141i)14-s + (−0.853 + 0.310i)16-s − 0.0529·17-s − 0.680i·19-s + (−0.706 − 0.592i)20-s + (−0.0131 − 0.0747i)22-s + (−0.551 + 1.51i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0318 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0318 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.0318 - 0.999i$
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: \((2,\ 567,\ (\ :1/2),\ -0.0318 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.994504 + 1.02675i\)
\(L(\frac12)\) \(\approx\) \(0.994504 + 1.02675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.158i)T \)
good2 \( 1 + (-0.159 + 0.190i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.62 - 1.36i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-0.919 + 1.09i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.30 - 3.59i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 0.218T + 17T^{2} \)
19 \( 1 + 2.96iT - 19T^{2} \)
23 \( 1 + (2.64 - 7.26i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.78 - 4.90i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.81 + 0.672i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (3.00 - 5.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.0 + 3.66i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.29 + 7.34i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.01 + 5.78i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-12.1 - 6.98i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.48 - 0.540i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-4.18 - 0.738i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-11.4 + 9.58i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-4.29 + 2.47i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.13 - 1.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.98 + 2.50i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.141 - 0.0515i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 6.80T + 89T^{2} \)
97 \( 1 + (12.2 + 2.16i)T + (91.1 + 33.1i)T^{2} \)
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

−11.41332175745570431592197571660, −10.31436549259709616564085039726, −8.949322441250541163787230582707, −8.290062148943567953593238911748, −7.28279857350507580837260283389, −6.89349418440019072020963389967, −5.20935359729699647813720269304, −4.13374114839759519202768518568, −3.33331612993047919628330991398, −1.96401658915580239864627798074, 0.815437706579017695181028003545, 2.26976090035011759367709293961, 4.17287440046851700742130195004, 4.86625577668928076200894533959, 5.78570815999285983203188974402, 6.88823632060125117882870364738, 8.050502522961099147575558965612, 8.482386484387413307954078156421, 9.876300583802976127684571213799, 10.42459503319338238756772610958

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