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Properties

Label 2-189-189.131-c1-0-2
Degree $2$
Conductor $189$
Sign $-0.965 - 0.260i$
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  + (1.61 + 1.92i)2-s + (−1.49 + 0.878i)3-s + (−0.750 + 4.25i)4-s + (−2.10 − 1.76i)5-s + (−4.10 − 1.45i)6-s + (−0.122 + 2.64i)7-s + (−5.05 + 2.91i)8-s + (1.45 − 2.62i)9-s − 6.90i·10-s + (2.69 + 3.20i)11-s + (−2.61 − 7.00i)12-s + (−0.0398 − 0.109i)13-s + (−5.28 + 4.03i)14-s + (4.69 + 0.786i)15-s + (−5.65 − 2.05i)16-s + 5.14·17-s + ⋯
L(s)  = 1  + (1.14 + 1.36i)2-s + (−0.861 + 0.507i)3-s + (−0.375 + 2.12i)4-s + (−0.940 − 0.789i)5-s + (−1.67 − 0.593i)6-s + (−0.0463 + 0.998i)7-s + (−1.78 + 1.03i)8-s + (0.485 − 0.874i)9-s − 2.18i·10-s + (0.811 + 0.966i)11-s + (−0.755 − 2.02i)12-s + (−0.0110 − 0.0303i)13-s + (−1.41 + 1.07i)14-s + (1.21 + 0.202i)15-s + (−1.41 − 0.514i)16-s + 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.965 - 0.260i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.965 - 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.178864 + 1.34715i\)
\(L(\frac12)\) \(\approx\) \(0.178864 + 1.34715i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.49 - 0.878i)T \)
7 \( 1 + (0.122 - 2.64i)T \)
good2 \( 1 + (-1.61 - 1.92i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (2.10 + 1.76i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (-2.69 - 3.20i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.0398 + 0.109i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 - 5.14T + 17T^{2} \)
19 \( 1 - 5.21iT - 19T^{2} \)
23 \( 1 + (2.56 + 7.05i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.21 + 3.34i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.305 + 0.0538i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (0.959 + 1.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.29 + 2.29i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.411 - 2.33i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.876 + 4.97i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (3.76 - 2.17i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.84 + 3.58i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.13 - 0.200i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-8.49 - 7.12i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (5.47 + 3.15i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.23 + 1.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12.0 + 10.1i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (10.6 + 3.88i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + 0.913T + 89T^{2} \)
97 \( 1 + (5.88 - 1.03i)T + (91.1 - 33.1i)T^{2} \)
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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

−12.62110626705629603403300356779, −12.31807341265942882971251040961, −11.78024881880612958704566808079, −9.930683487748627443532017703976, −8.641576080176603847263255356075, −7.69580352981189992806790526699, −6.42817225102670145213563355586, −5.57319882616296115456264903072, −4.58036220051958169359503835471, −3.81961180332311301157886471575, 1.07463773182493354436306605014, 3.22127902250597008593504479968, 4.08582595493178479288428070155, 5.43144946122959361557943801119, 6.65644890309340512483265770120, 7.69498516417435951469110864009, 9.781734423199390487804288298279, 10.85565469614595862614416883082, 11.30062327826848593909303349276, 11.88731903517431885065303708958

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