↑

Properties

Label 2-189-7.6-c2-0-11
Degree $2$
Conductor $189$
Sign $-0.452 + 0.891i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.66·2-s − 1.24·4-s + 6.94i·5-s + (−6.24 − 3.16i)7-s + 8.70·8-s − 11.5i·10-s − 1.66·11-s − 11.4i·13-s + (10.3 + 5.25i)14-s − 9.48·16-s − 12.2i·17-s − 12.5i·19-s − 8.62i·20-s + 2.75·22-s − 39.8·23-s + ⋯
L(s)  = 1  − 0.830·2-s − 0.310·4-s + 1.38i·5-s + (−0.891 − 0.452i)7-s + 1.08·8-s − 1.15i·10-s − 0.150·11-s − 0.877i·13-s + (0.740 + 0.375i)14-s − 0.592·16-s − 0.717i·17-s − 0.660i·19-s − 0.431i·20-s + 0.125·22-s − 1.73·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.452 + 0.891i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ -0.452 + 0.891i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.135225 - 0.220230i\)
\(L(\frac12)\) \(\approx\) \(0.135225 - 0.220230i\)
\(L(2)\) 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 \( 1 + (6.24 + 3.16i)T \)
good2 \( 1 + 1.66T + 4T^{2} \)
5 \( 1 - 6.94iT - 25T^{2} \)
11 \( 1 + 1.66T + 121T^{2} \)
13 \( 1 + 11.4iT - 169T^{2} \)
17 \( 1 + 12.2iT - 289T^{2} \)
19 \( 1 + 12.5iT - 361T^{2} \)
23 \( 1 + 39.8T + 529T^{2} \)
29 \( 1 - 6.23T + 841T^{2} \)
31 \( 1 + 50.3iT - 961T^{2} \)
37 \( 1 - 20.5T + 1.36e3T^{2} \)
41 \( 1 + 69.2iT - 1.68e3T^{2} \)
43 \( 1 + 51.6T + 1.84e3T^{2} \)
47 \( 1 - 36.1iT - 2.20e3T^{2} \)
53 \( 1 - 37.3T + 2.80e3T^{2} \)
59 \( 1 - 98.8iT - 3.48e3T^{2} \)
61 \( 1 - 73.2iT - 3.72e3T^{2} \)
67 \( 1 + 11.1T + 4.48e3T^{2} \)
71 \( 1 + 130.T + 5.04e3T^{2} \)
73 \( 1 + 94.7iT - 5.32e3T^{2} \)
79 \( 1 - 9.75T + 6.24e3T^{2} \)
83 \( 1 + 91.7iT - 6.88e3T^{2} \)
89 \( 1 - 104. iT - 7.92e3T^{2} \)
97 \( 1 + 52.9iT - 9.40e3T^{2} \)
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

−11.78472804882254344423291273906, −10.51921857893501781577850895942, −10.19118743359255976142867762958, −9.203350888790776075259310987257, −7.81287719788936726186871266584, −7.12408896612671654352106829630, −5.90019016719042191824692148904, −4.06646174928839419728605502913, −2.71165015861368422805027733163, −0.20414321394762066302837105153, 1.58396037256154199187561360575, 3.95898060011725574172491546352, 5.12083249412495719243033035698, 6.44829286177956703280263062756, 8.088317084703007417600371383659, 8.658025772220451731987327940467, 9.574108155052720242270845782928, 10.21255908411176611186732555667, 11.82687665262172091434164241318, 12.65012100517238393452775816760

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