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Properties

Label 2-189-63.31-c2-0-8
Degree $2$
Conductor $189$
Sign $0.985 - 0.167i$
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  + (0.227 − 0.394i)2-s + (1.89 + 3.28i)4-s − 4.37i·5-s + (5.22 + 4.66i)7-s + 3.54·8-s + (−1.72 − 0.994i)10-s + 0.139·11-s + (−1.71 − 0.987i)13-s + (3.02 − 0.996i)14-s + (−6.77 + 11.7i)16-s + (26.7 + 15.4i)17-s + (25.2 − 14.5i)19-s + (14.3 − 8.28i)20-s + (0.0317 − 0.0549i)22-s − 29.7·23-s + ⋯
L(s)  = 1  + (0.113 − 0.197i)2-s + (0.474 + 0.821i)4-s − 0.874i·5-s + (0.745 + 0.666i)7-s + 0.443·8-s + (−0.172 − 0.0994i)10-s + 0.0126·11-s + (−0.131 − 0.0759i)13-s + (0.216 − 0.0711i)14-s + (−0.423 + 0.733i)16-s + (1.57 + 0.909i)17-s + (1.32 − 0.766i)19-s + (0.717 − 0.414i)20-s + (0.00144 − 0.00249i)22-s − 1.29·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.985 - 0.167i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.985 - 0.167i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.93856 + 0.163139i\)
\(L(\frac12)\) \(\approx\) \(1.93856 + 0.163139i\)
\(L(2)\) 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 \( 1 + (-5.22 - 4.66i)T \)
good2 \( 1 + (-0.227 + 0.394i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + 4.37iT - 25T^{2} \)
11 \( 1 - 0.139T + 121T^{2} \)
13 \( 1 + (1.71 + 0.987i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-26.7 - 15.4i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-25.2 + 14.5i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + 29.7T + 529T^{2} \)
29 \( 1 + (7.28 + 12.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (6.82 - 3.94i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (7.73 + 13.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (0.747 + 0.431i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (15.6 + 27.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (58.0 + 33.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (16.9 - 29.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (57.4 - 33.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (35.9 + 20.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (51.7 + 89.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 86.4T + 5.04e3T^{2} \)
73 \( 1 + (28.6 + 16.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (24.3 - 42.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-102. + 59.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-33.1 + 19.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-70.3 + 40.6i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.08537960334559842787032847187, −11.85431712018919145443101605555, −10.53072442798740130807820210189, −9.217579977757203422261459628856, −8.196999281266310388677502802758, −7.54857144885217937170389355902, −5.85723408411148969986098180075, −4.75234326202212786153034471676, −3.31433215157830851763431372029, −1.68505631271961876443454277244, 1.42405510765380279524930645753, 3.21712443933130922970079805856, 4.92861902279437934701194123689, 5.98833108945705592797396712928, 7.22569149077183220353617958276, 7.81201767847550637223406804492, 9.743749240202873661612914446537, 10.28487463599692479408991569343, 11.30236900126127497911737141724, 11.99609163691948039671059680162

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