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Properties

Label 2-189-189.131-c1-0-21
Degree $2$
Conductor $189$
Sign $-0.451 - 0.892i$
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  + (−1.44 − 1.72i)2-s + (0.462 − 1.66i)3-s + (−0.536 + 3.03i)4-s + (−2.61 − 2.19i)5-s + (−3.55 + 1.61i)6-s + (−0.126 + 2.64i)7-s + (2.12 − 1.22i)8-s + (−2.57 − 1.54i)9-s + 7.69i·10-s + (1.63 + 1.94i)11-s + (4.82 + 2.30i)12-s + (−2.11 − 5.80i)13-s + (4.74 − 3.61i)14-s + (−4.87 + 3.34i)15-s + (0.606 + 0.220i)16-s − 2.39·17-s + ⋯
L(s)  = 1  + (−1.02 − 1.22i)2-s + (0.267 − 0.963i)3-s + (−0.268 + 1.51i)4-s + (−1.16 − 0.981i)5-s + (−1.45 + 0.661i)6-s + (−0.0479 + 0.998i)7-s + (0.750 − 0.433i)8-s + (−0.857 − 0.515i)9-s + 2.43i·10-s + (0.493 + 0.587i)11-s + (1.39 + 0.664i)12-s + (−0.585 − 1.60i)13-s + (1.26 − 0.965i)14-s + (−1.25 + 0.864i)15-s + (0.151 + 0.0551i)16-s − 0.581·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.451 - 0.892i$
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: \((2,\ 189,\ (\ :1/2),\ -0.451 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185342 + 0.301386i\)
\(L(\frac12)\) \(\approx\) \(0.185342 + 0.301386i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.462 + 1.66i)T \)
7 \( 1 + (0.126 - 2.64i)T \)
good2 \( 1 + (1.44 + 1.72i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (2.61 + 2.19i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (-1.63 - 1.94i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (2.11 + 5.80i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + 2.39T + 17T^{2} \)
19 \( 1 + 1.24iT - 19T^{2} \)
23 \( 1 + (-0.769 - 2.11i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.685 + 1.88i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (1.55 + 0.273i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (5.01 + 8.68i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.39 - 1.96i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.110 + 0.625i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (2.09 + 11.8i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-4.15 + 2.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.59 + 0.943i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-7.26 + 1.28i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.14 + 5.15i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-4.45 - 2.57i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-11.6 - 6.70i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.740 + 0.621i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (2.38 + 0.867i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + 9.95T + 89T^{2} \)
97 \( 1 + (-7.93 + 1.39i)T + (91.1 - 33.1i)T^{2} \)
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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

−11.99211197629283683038134563511, −11.22423646190872112756701357462, −9.758904362228302753935700940256, −8.727661304899111830144219386825, −8.297222988536945346197159604520, −7.27037609134498607113898207338, −5.35422702568596989084224253407, −3.47510959048939733950330231389, −2.11665043212231703650988147600, −0.41681279251393353293675162520, 3.48161052182281085965617564314, 4.57091936113509169226786747300, 6.50821582957027958005838695455, 7.13947638032711984236088538948, 8.171907004942226047954782572379, 9.058269193106664027283124084607, 10.07794905559046787175574263705, 10.93783120565391799753628769026, 11.76660857460053654694845263506, 13.96188046545463613954963042397

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