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Properties

Label 2-189-63.47-c3-0-0
Degree $2$
Conductor $189$
Sign $-0.759 + 0.651i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.647 + 0.373i)2-s + (−3.72 + 6.44i)4-s − 8.70·5-s + (18.2 + 2.85i)7-s − 11.5i·8-s + (5.63 − 3.25i)10-s + 41.5i·11-s + (−33.0 + 19.0i)13-s + (−12.9 + 4.99i)14-s + (−25.4 − 44.0i)16-s + (−39.9 − 69.2i)17-s + (−49.4 − 28.5i)19-s + (32.3 − 56.0i)20-s + (−15.5 − 26.8i)22-s − 177. i·23-s + ⋯
L(s)  = 1  + (−0.228 + 0.132i)2-s + (−0.465 + 0.805i)4-s − 0.778·5-s + (0.988 + 0.154i)7-s − 0.510i·8-s + (0.178 − 0.102i)10-s + 1.13i·11-s + (−0.704 + 0.406i)13-s + (−0.246 + 0.0953i)14-s + (−0.397 − 0.688i)16-s + (−0.570 − 0.987i)17-s + (−0.596 − 0.344i)19-s + (0.361 − 0.626i)20-s + (−0.150 − 0.260i)22-s − 1.60i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.651i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.759 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.759 + 0.651i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.759 + 0.651i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0122307 - 0.0330489i\)
\(L(\frac12)\) \(\approx\) \(0.0122307 - 0.0330489i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.2 - 2.85i)T \)
good2 \( 1 + (0.647 - 0.373i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 8.70T + 125T^{2} \)
11 \( 1 - 41.5iT - 1.33e3T^{2} \)
13 \( 1 + (33.0 - 19.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (39.9 + 69.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (49.4 + 28.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 177. iT - 1.21e4T^{2} \)
29 \( 1 + (-60.1 - 34.7i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (225. + 130. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (74.9 - 129. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-54.7 - 94.7i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-124. + 215. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (147. + 255. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (263. - 152. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (360. - 624. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (561. - 324. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-97.4 + 168. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 98.7iT - 3.57e5T^{2} \)
73 \( 1 + (-226. + 130. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-577. - 9.99e2i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-285. + 494. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (89.5 - 155. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (956. + 552. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.40143038622940127107125567045, −11.96582028422297896682779074207, −10.86932297999937461370923362311, −9.487622071185100748144985899757, −8.600571309678080425341824873723, −7.64170782759244161092424508827, −6.95439604561284944413682905119, −4.79385166734255476350229376527, −4.24456755438064245057621777219, −2.39637367725232490279675715101, 0.01687242934419995086997979757, 1.64802389768111855394804924350, 3.75741485289826530302123460326, 4.97806771823332197977982787037, 6.02599820254416739343454907768, 7.70252726252227576435556457663, 8.410429593351266105259435464563, 9.473879736954786902232195523930, 10.87180494525554147774649030556, 11.08216629760820789487570782412

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