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Properties

Label 2-189-7.2-c3-0-18
Degree $2$
Conductor $189$
Sign $0.843 + 0.537i$
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  + (1.74 − 3.02i)2-s + (−2.09 − 3.63i)4-s + (−3.93 + 6.82i)5-s + (18.4 + 1.98i)7-s + 13.2·8-s + (13.7 + 23.8i)10-s + (27.0 + 46.7i)11-s − 48.9·13-s + (38.1 − 52.2i)14-s + (39.9 − 69.2i)16-s + (48.4 + 83.9i)17-s + (71.1 − 123. i)19-s + 33.0·20-s + 188.·22-s + (51.8 − 89.8i)23-s + ⋯
L(s)  = 1  + (0.617 − 1.06i)2-s + (−0.262 − 0.454i)4-s + (−0.352 + 0.610i)5-s + (0.994 + 0.107i)7-s + 0.587·8-s + (0.434 + 0.753i)10-s + (0.740 + 1.28i)11-s − 1.04·13-s + (0.728 − 0.996i)14-s + (0.624 − 1.08i)16-s + (0.691 + 1.19i)17-s + (0.859 − 1.48i)19-s + 0.369·20-s + 1.82·22-s + (0.470 − 0.814i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.843 + 0.537i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.843 + 0.537i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.60637 - 0.760310i\)
\(L(\frac12)\) \(\approx\) \(2.60637 - 0.760310i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.4 - 1.98i)T \)
good2 \( 1 + (-1.74 + 3.02i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (3.93 - 6.82i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-27.0 - 46.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 48.9T + 2.19e3T^{2} \)
17 \( 1 + (-48.4 - 83.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-71.1 + 123. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-51.8 + 89.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 24.5T + 2.43e4T^{2} \)
31 \( 1 + (93.6 + 162. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (73.2 - 126. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 314.T + 6.89e4T^{2} \)
43 \( 1 - 173.T + 7.95e4T^{2} \)
47 \( 1 + (129. - 224. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-310. - 537. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (221. + 383. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (56.6 - 98.1i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (314. + 544. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 41.3T + 3.57e5T^{2} \)
73 \( 1 + (223. + 387. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (217. - 376. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 329.T + 5.71e5T^{2} \)
89 \( 1 + (12.4 - 21.5i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 499.T + 9.12e5T^{2} \)
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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

−12.00801819665535874526240581261, −11.23054881887706513308954713807, −10.42745121455488139383936100620, −9.333421204763229046345425792743, −7.73436624219021573348330776913, −7.01789404072123880993653355558, −5.06300188908236063834887638600, −4.22713389847291686029223558723, −2.83864365127540463322230083519, −1.60425677198612484858862992372, 1.22190223810963536242845293610, 3.66718340543381395435956028114, 5.02380435181217762435983764295, 5.56805896914625300196554581247, 7.09619019279277253095947372198, 7.85633057654771731854772652099, 8.825342100120481229566811566819, 10.21347727622361647322256792934, 11.53412653189414083225385237081, 12.16631361105984067784607048792

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