Properties

Label 2-189-189.47-c1-0-15
Degree $2$
Conductor $189$
Sign $0.935 + 0.353i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.882 + 0.155i)2-s + (1.65 − 0.495i)3-s + (−1.12 − 0.409i)4-s + (0.123 + 0.0449i)5-s + (1.54 − 0.178i)6-s + (1.69 − 2.02i)7-s + (−2.48 − 1.43i)8-s + (2.50 − 1.64i)9-s + (0.101 + 0.0588i)10-s + (1.37 + 3.76i)11-s + (−2.07 − 0.122i)12-s + (−1.02 + 2.81i)13-s + (1.81 − 1.52i)14-s + (0.227 + 0.0134i)15-s + (−0.130 − 0.109i)16-s + (−0.172 + 0.298i)17-s + ⋯
L(s)  = 1  + (0.623 + 0.109i)2-s + (0.958 − 0.286i)3-s + (−0.562 − 0.204i)4-s + (0.0552 + 0.0201i)5-s + (0.629 − 0.0730i)6-s + (0.642 − 0.766i)7-s + (−0.877 − 0.506i)8-s + (0.836 − 0.548i)9-s + (0.0322 + 0.0186i)10-s + (0.413 + 1.13i)11-s + (−0.597 − 0.0352i)12-s + (−0.284 + 0.781i)13-s + (0.484 − 0.407i)14-s + (0.0586 + 0.00346i)15-s + (−0.0326 − 0.0274i)16-s + (−0.0417 + 0.0723i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.935 + 0.353i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.935 + 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81773 - 0.331707i\)
\(L(\frac12)\) \(\approx\) \(1.81773 - 0.331707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 + 0.495i)T \)
7 \( 1 + (-1.69 + 2.02i)T \)
good2 \( 1 + (-0.882 - 0.155i)T + (1.87 + 0.684i)T^{2} \)
5 \( 1 + (-0.123 - 0.0449i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (-1.37 - 3.76i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (1.02 - 2.81i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.172 - 0.298i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.24 - 2.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.27 - 1.10i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.29 - 6.31i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.98 + 8.20i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + 7.28T + 37T^{2} \)
41 \( 1 + (-9.04 - 3.29i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.350 + 1.98i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-4.17 + 1.52i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (5.49 - 3.17i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.136 + 0.114i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.59 + 4.38i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.00 - 11.3i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-0.373 + 0.215i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.52iT - 73T^{2} \)
79 \( 1 + (-0.410 + 2.32i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-6.81 + 2.48i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (7.70 + 13.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.2 - 2.15i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.68739518638277663301033313682, −11.95950451349650464757620339481, −10.25519080215104073173069728639, −9.539599455191411100001358005321, −8.437321950322206136796705163106, −7.39178715250901920376642841058, −6.27045005929313687875602899682, −4.47058159591585204609627421673, −3.99103022789796773953379393674, −1.91539112346061206793574047575, 2.51112055049641946326149327704, 3.72960770245025039523996421199, 4.87111013009300939021522093560, 6.01885687752001932535868617123, 7.967816612220842834433022035293, 8.561210377502070584248430164645, 9.385337636452348743284339709056, 10.68385493131811352453651422036, 11.91611879197219430239546637895, 12.75596778966437475044247450397

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