↑

Properties

Label 2-189-9.4-c3-0-14
Degree $2$
Conductor $189$
Sign $-0.606 - 0.795i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.47 − 4.28i)2-s + (−8.26 + 14.3i)4-s + (8.06 − 13.9i)5-s + (3.5 + 6.06i)7-s + 42.1·8-s − 79.9·10-s + (−15.3 − 26.5i)11-s + (39.0 − 67.5i)13-s + (17.3 − 30.0i)14-s + (−38.3 − 66.4i)16-s − 106.·17-s − 49.5·19-s + (133. + 230. i)20-s + (−75.8 + 131. i)22-s + (19.5 − 33.7i)23-s + ⋯
L(s)  = 1  + (−0.875 − 1.51i)2-s + (−1.03 + 1.78i)4-s + (0.721 − 1.25i)5-s + (0.188 + 0.327i)7-s + 1.86·8-s − 2.52·10-s + (−0.419 − 0.727i)11-s + (0.832 − 1.44i)13-s + (0.330 − 0.573i)14-s + (−0.599 − 1.03i)16-s − 1.52·17-s − 0.597·19-s + (1.49 + 2.58i)20-s + (−0.735 + 1.27i)22-s + (0.176 − 0.306i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.606 - 0.795i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.606 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.334104 + 0.674788i\)
\(L(\frac12)\) \(\approx\) \(0.334104 + 0.674788i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.5 - 6.06i)T \)
good2 \( 1 + (2.47 + 4.28i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-8.06 + 13.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (15.3 + 26.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-39.0 + 67.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 106.T + 4.91e3T^{2} \)
19 \( 1 + 49.5T + 6.85e3T^{2} \)
23 \( 1 + (-19.5 + 33.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-2.87 - 4.97i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (92.2 - 159. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 91.0T + 5.06e4T^{2} \)
41 \( 1 + (41.4 - 71.7i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-45.6 - 79.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-118. - 206. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 497.T + 1.48e5T^{2} \)
59 \( 1 + (2.10 - 3.64i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (313. + 542. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-339. + 587. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 747.T + 3.57e5T^{2} \)
73 \( 1 + 23.2T + 3.89e5T^{2} \)
79 \( 1 + (77.0 + 133. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-141. - 244. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 111.T + 7.04e5T^{2} \)
97 \( 1 + (555. + 961. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−11.13474808090228426821142367878, −10.66608688167902053942916231797, −9.455431418229349819067549793689, −8.674414989304300615359486033273, −8.192354386400481141185029041597, −5.94870726882154060831422391446, −4.65526310634522306321658542524, −3.04087054131559723535364000085, −1.69121384978491797765666514760, −0.44161268023358433569670461834, 2.03123781982757143912476870933, 4.39733999202607256557329696120, 5.96928562632231691362643342749, 6.72984418754301920033722628425, 7.33294545996766760447335019218, 8.687018618128096466935605867305, 9.517314564135880027820110227848, 10.48274816056075564469724974210, 11.29414263355665862191947804528, 13.30048023253634638497535840541

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