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Properties

Label 2-189-63.41-c3-0-0
Degree $2$
Conductor $189$
Sign $-0.999 - 0.0186i$
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  + (0.0847 − 0.0489i)2-s + (−3.99 + 6.91i)4-s + (9.06 − 15.6i)5-s + (−18.0 + 4.32i)7-s + 1.56i·8-s − 1.77i·10-s + (−32.0 + 18.5i)11-s + (16.3 + 9.44i)13-s + (−1.31 + 1.24i)14-s + (−31.8 − 55.2i)16-s − 62.5·17-s + 70.2i·19-s + (72.4 + 125. i)20-s + (−1.81 + 3.13i)22-s + (−140. − 81.2i)23-s + ⋯
L(s)  = 1  + (0.0299 − 0.0173i)2-s + (−0.499 + 0.864i)4-s + (0.810 − 1.40i)5-s + (−0.972 + 0.233i)7-s + 0.0691i·8-s − 0.0561i·10-s + (−0.878 + 0.507i)11-s + (0.349 + 0.201i)13-s + (−0.0251 + 0.0238i)14-s + (−0.498 − 0.862i)16-s − 0.891·17-s + 0.848i·19-s + (0.809 + 1.40i)20-s + (−0.0175 + 0.0304i)22-s + (−1.27 − 0.736i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0186i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.999 - 0.0186i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.999 - 0.0186i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.000167298 + 0.0179524i\)
\(L(\frac12)\) \(\approx\) \(0.000167298 + 0.0179524i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.0 - 4.32i)T \)
good2 \( 1 + (-0.0847 + 0.0489i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-9.06 + 15.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (32.0 - 18.5i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-16.3 - 9.44i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 62.5T + 4.91e3T^{2} \)
19 \( 1 - 70.2iT - 6.85e3T^{2} \)
23 \( 1 + (140. + 81.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (82.5 - 47.6i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (110. + 63.8i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 378.T + 5.06e4T^{2} \)
41 \( 1 + (-99.1 + 171. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-160. - 278. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-79.3 - 137. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 191. iT - 1.48e5T^{2} \)
59 \( 1 + (106. - 185. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-190. + 110. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-68.2 + 118. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 458. iT - 3.57e5T^{2} \)
73 \( 1 + 967. iT - 3.89e5T^{2} \)
79 \( 1 + (-298. - 516. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-180. - 313. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 35.4T + 7.04e5T^{2} \)
97 \( 1 + (-1.15e3 + 664. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.67565996254812882759545498762, −12.14833439776399321267373644882, −10.39539502941963037218820017177, −9.350153789758548032318925678388, −8.792851409634041208097677944161, −7.71727986559736749667277179116, −6.16842257509329659979274360175, −5.04775668065536603552260681926, −3.90415883416517977156480728521, −2.17983786310297187007683012689, 0.00726826953917293036107539182, 2.26216091729773417225421807330, 3.61959590980410383966677492558, 5.46785604789830450073349767718, 6.25663998807800341726132589839, 7.16056562582780753534896183891, 8.864410846680777783467675601249, 9.897503818265645527063033529806, 10.47164917671756269573496874756, 11.21360668349176482700517700089

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