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Properties

Label 2-189-189.88-c1-0-7
Degree $2$
Conductor $189$
Sign $-0.0302 + 0.999i$
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  + (−1.86 − 1.56i)2-s + (1.55 − 0.761i)3-s + (0.681 + 3.86i)4-s + (1.94 − 1.63i)5-s + (−4.09 − 1.01i)6-s + (1.59 + 2.11i)7-s + (2.34 − 4.05i)8-s + (1.84 − 2.36i)9-s − 6.18·10-s + (2.75 + 2.31i)11-s + (4.00 + 5.49i)12-s + (0.574 + 0.209i)13-s + (0.334 − 6.43i)14-s + (1.78 − 4.02i)15-s + (−3.33 + 1.21i)16-s − 2.17·17-s + ⋯
L(s)  = 1  + (−1.31 − 1.10i)2-s + (0.898 − 0.439i)3-s + (0.340 + 1.93i)4-s + (0.870 − 0.730i)5-s + (−1.67 − 0.414i)6-s + (0.602 + 0.798i)7-s + (0.827 − 1.43i)8-s + (0.613 − 0.789i)9-s − 1.95·10-s + (0.832 + 0.698i)11-s + (1.15 + 1.58i)12-s + (0.159 + 0.0579i)13-s + (0.0893 − 1.71i)14-s + (0.461 − 1.03i)15-s + (−0.833 + 0.303i)16-s − 0.528·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0302 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0302 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.0302 + 0.999i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.0302 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.687355 - 0.708501i\)
\(L(\frac12)\) \(\approx\) \(0.687355 - 0.708501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.55 + 0.761i)T \)
7 \( 1 + (-1.59 - 2.11i)T \)
good2 \( 1 + (1.86 + 1.56i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (-1.94 + 1.63i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-2.75 - 2.31i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.574 - 0.209i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + 2.17T + 17T^{2} \)
19 \( 1 + 6.48T + 19T^{2} \)
23 \( 1 + (8.52 + 3.10i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (2.61 - 0.953i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-0.599 - 3.40i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-4.59 + 7.96i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.95 - 0.711i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.636 - 3.60i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.15 - 6.56i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-1.94 + 3.36i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.14 + 1.50i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.758 - 4.30i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-7.33 + 6.15i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.53 - 2.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.73 + 11.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.81 - 6.55i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.25 - 0.456i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + (-0.474 + 2.68i)T + (-91.1 - 33.1i)T^{2} \)
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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

−12.45387770169015805360234448061, −11.25838884503847114916547994020, −10.00971850800966773260915791866, −9.182323609443255227492667559462, −8.725917645031138012863617523277, −7.83233727219040632702037722647, −6.27043508451458319242626507663, −4.21700200635247344820808338721, −2.28212966841650576218656206667, −1.68251409212076582302026452569, 1.91564033930307733081465068876, 4.06218021384716147687206616824, 5.95899913555707891653575358814, 6.82872593452336073750171209324, 7.967450550830096394238167949142, 8.650219401588226128608489898626, 9.734140018745628278436117750472, 10.31249604472272920287796743066, 11.20173981825495699526741866952, 13.45674302477229975564182674380

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