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Properties

Label 2-189-7.6-c2-0-10
Degree $2$
Conductor $189$
Sign $0.565 - 0.824i$
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  + 2.28·2-s + 1.22·4-s + 6.53i·5-s + (5.77 + 3.96i)7-s − 6.33·8-s + 14.9i·10-s + 13.1·11-s + 3.96i·13-s + (13.1 + 9.05i)14-s − 19.4·16-s + 15.5i·17-s − 30.7i·19-s + 8.02i·20-s + 30.1·22-s − 1.76·23-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.306·4-s + 1.30i·5-s + (0.824 + 0.565i)7-s − 0.792·8-s + 1.49i·10-s + 1.19·11-s + 0.304i·13-s + (0.942 + 0.646i)14-s − 1.21·16-s + 0.916i·17-s − 1.61i·19-s + 0.401i·20-s + 1.37·22-s − 0.0767·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.565 - 0.824i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.565 - 0.824i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.28761 + 1.20471i\)
\(L(\frac12)\) \(\approx\) \(2.28761 + 1.20471i\)
\(L(2)\) 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 \( 1 + (-5.77 - 3.96i)T \)
good2 \( 1 - 2.28T + 4T^{2} \)
5 \( 1 - 6.53iT - 25T^{2} \)
11 \( 1 - 13.1T + 121T^{2} \)
13 \( 1 - 3.96iT - 169T^{2} \)
17 \( 1 - 15.5iT - 289T^{2} \)
19 \( 1 + 30.7iT - 361T^{2} \)
23 \( 1 + 1.76T + 529T^{2} \)
29 \( 1 - 37.8T + 841T^{2} \)
31 \( 1 + 57.5iT - 961T^{2} \)
37 \( 1 + 41.7T + 1.36e3T^{2} \)
41 \( 1 + 24.6iT - 1.68e3T^{2} \)
43 \( 1 + 38.0T + 1.84e3T^{2} \)
47 \( 1 - 32.6iT - 2.20e3T^{2} \)
53 \( 1 + 7.38T + 2.80e3T^{2} \)
59 \( 1 + 59.8iT - 3.48e3T^{2} \)
61 \( 1 + 32.5iT - 3.72e3T^{2} \)
67 \( 1 - 91.5T + 4.48e3T^{2} \)
71 \( 1 + 9.86T + 5.04e3T^{2} \)
73 \( 1 + 23.7iT - 5.32e3T^{2} \)
79 \( 1 - 74.6T + 6.24e3T^{2} \)
83 \( 1 - 126. iT - 6.88e3T^{2} \)
89 \( 1 - 27.1iT - 7.92e3T^{2} \)
97 \( 1 - 48.4iT - 9.40e3T^{2} \)
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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

−12.47567328080158320089379059734, −11.56829129468189744712131005650, −10.99779126951729052756311430434, −9.510927735492170456599749678543, −8.448815030393598924448779492403, −6.86592498155258178202539984945, −6.16747562585706064339255148684, −4.81724624922673939771273302049, −3.69249376361282878030848787359, −2.38207863132079158455290062966, 1.27450311124037283447043395213, 3.61077683795664290587029378568, 4.64962613380902747133053085015, 5.32955661100987159163360262556, 6.71151061663366347619229803858, 8.257036702729933310227361267078, 9.003765397188964744176535237877, 10.27929579580757851772392701408, 11.95415633467052493654545948113, 12.04426467934866630023979411566

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