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Properties

Label 2-189-63.61-c2-0-2
Degree $2$
Conductor $189$
Sign $-0.981 - 0.189i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.67 + 2.90i)2-s + (−3.62 + 6.28i)4-s + 0.888i·5-s + (−3.29 + 6.17i)7-s − 10.9·8-s + (−2.58 + 1.49i)10-s + 5.50·11-s + (−12.3 + 7.11i)13-s + (−23.4 + 0.792i)14-s + (−3.81 − 6.61i)16-s + (11.2 − 6.51i)17-s + (−22.7 − 13.1i)19-s + (−5.58 − 3.22i)20-s + (9.23 + 16.0i)22-s + 36.0·23-s + ⋯
L(s)  = 1  + (0.838 + 1.45i)2-s + (−0.907 + 1.57i)4-s + 0.177i·5-s + (−0.470 + 0.882i)7-s − 1.36·8-s + (−0.258 + 0.149i)10-s + 0.500·11-s + (−0.948 + 0.547i)13-s + (−1.67 + 0.0566i)14-s + (−0.238 − 0.413i)16-s + (0.664 − 0.383i)17-s + (−1.19 − 0.690i)19-s + (−0.279 − 0.161i)20-s + (0.419 + 0.727i)22-s + 1.56·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.981 - 0.189i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ -0.981 - 0.189i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.190584 + 1.99543i\)
\(L(\frac12)\) \(\approx\) \(0.190584 + 1.99543i\)
\(L(2)\) 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 \( 1 + (3.29 - 6.17i)T \)
good2 \( 1 + (-1.67 - 2.90i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 - 0.888iT - 25T^{2} \)
11 \( 1 - 5.50T + 121T^{2} \)
13 \( 1 + (12.3 - 7.11i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-11.2 + 6.51i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (22.7 + 13.1i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 - 36.0T + 529T^{2} \)
29 \( 1 + (16.4 - 28.5i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-32.8 - 18.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-21.4 + 37.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-51.1 + 29.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (14.4 - 25.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-10.8 + 6.27i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-4.88 - 8.45i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (21.3 + 12.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.7 - 9.08i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.1 - 26.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 43.4T + 5.04e3T^{2} \)
73 \( 1 + (-23.5 + 13.6i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-33.7 - 58.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-8.30 - 4.79i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (54.8 + 31.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (76.9 + 44.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.81250732672575829814188231175, −12.38792895505990215791039269192, −10.99512311623181533377829965312, −9.369100682947073511576202581789, −8.632688101197147943206345328140, −7.19922661484645652612375716310, −6.62335643623426532037365093774, −5.45417847087443495379847141966, −4.50386922427814362450066469574, −2.90967927045836332882834587566, 0.972309915953531180431083759645, 2.73459806145355769518785896675, 3.92813936371698117874396790314, 4.88674630799738462985698026045, 6.33624960017093720873222503073, 7.80404445976022495812949334863, 9.411233029523923181580221740184, 10.21989650157777028056803947031, 10.93694106473160598196140230671, 12.03161865877955691048035116287

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