↑

Properties

Label 2-189-63.47-c1-0-5
Degree $2$
Conductor $189$
Sign $0.330 + 0.943i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
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  + (2.24 − 1.29i)2-s + (2.36 − 4.09i)4-s − 1.25·5-s + (−0.648 + 2.56i)7-s − 7.07i·8-s + (−2.81 + 1.62i)10-s + 0.616i·11-s + (−1.06 + 0.613i)13-s + (1.87 + 6.60i)14-s + (−4.44 − 7.69i)16-s + (2.21 + 3.83i)17-s + (−1.64 − 0.950i)19-s + (−2.96 + 5.12i)20-s + (0.799 + 1.38i)22-s − 4.74i·23-s + ⋯
L(s)  = 1  + (1.58 − 0.916i)2-s + (1.18 − 2.04i)4-s − 0.560·5-s + (−0.245 + 0.969i)7-s − 2.50i·8-s + (−0.889 + 0.513i)10-s + 0.185i·11-s + (−0.294 + 0.170i)13-s + (0.499 + 1.76i)14-s + (−1.11 − 1.92i)16-s + (0.537 + 0.930i)17-s + (−0.377 − 0.218i)19-s + (−0.662 + 1.14i)20-s + (0.170 + 0.295i)22-s − 0.990i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.330 + 0.943i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.330 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93040 - 1.36889i\)
\(L(\frac12)\) \(\approx\) \(1.93040 - 1.36889i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.648 - 2.56i)T \)
good2 \( 1 + (-2.24 + 1.29i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 1.25T + 5T^{2} \)
11 \( 1 - 0.616iT - 11T^{2} \)
13 \( 1 + (1.06 - 0.613i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.21 - 3.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.64 + 0.950i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.74iT - 23T^{2} \)
29 \( 1 + (-5.07 - 2.93i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.14 + 1.24i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.33 + 2.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.09 + 3.63i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.24 - 3.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.80 + 6.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.67 - 1.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.78 - 3.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.5 + 7.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.80 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (-9.95 + 5.74i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.01 - 3.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.36 - 7.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.811 + 1.40i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.76 + 5.06i)T + (48.5 + 84.0i)T^{2} \)
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

−12.38838193132915271546474644155, −11.78423775574849767054745676372, −10.79824005238752210424487630366, −9.818192438208894785162445632454, −8.378202968708365846266869542646, −6.69131810032389726551135954267, −5.67973715468400867429754720813, −4.57382888118374017129140050805, −3.43304210968603055617101126965, −2.15634764867644485394192056760, 3.16196804717800642795217538994, 4.13049287557285688736249042805, 5.19566346007666920374338565973, 6.43883590422811750030284752483, 7.38944370554384841184678604437, 8.068482853697023154407209929959, 9.876406886503822901439938695624, 11.30652634770891341642356902983, 12.04632236495153400294574589707, 13.05180055924779381549725463683

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