Properties

Label 2-189-189.47-c1-0-13
Degree $2$
Conductor $189$
Sign $0.109 + 0.993i$
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  + (0.0159 + 0.00280i)2-s + (−0.402 − 1.68i)3-s + (−1.87 − 0.683i)4-s + (3.75 + 1.36i)5-s + (−0.00168 − 0.0279i)6-s + (0.157 − 2.64i)7-s + (−0.0559 − 0.0323i)8-s + (−2.67 + 1.35i)9-s + (0.0558 + 0.0322i)10-s + (−1.76 − 4.84i)11-s + (−0.395 + 3.44i)12-s + (−0.220 + 0.606i)13-s + (0.00992 − 0.0415i)14-s + (0.789 − 6.86i)15-s + (3.06 + 2.57i)16-s + (1.69 − 2.93i)17-s + ⋯
L(s)  = 1  + (0.0112 + 0.00198i)2-s + (−0.232 − 0.972i)3-s + (−0.939 − 0.341i)4-s + (1.67 + 0.610i)5-s + (−0.000686 − 0.0114i)6-s + (0.0596 − 0.998i)7-s + (−0.0197 − 0.0114i)8-s + (−0.891 + 0.452i)9-s + (0.0176 + 0.0102i)10-s + (−0.531 − 1.46i)11-s + (−0.114 + 0.993i)12-s + (−0.0612 + 0.168i)13-s + (0.00265 − 0.0111i)14-s + (0.203 − 1.77i)15-s + (0.765 + 0.642i)16-s + (0.410 − 0.711i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.109 + 0.993i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.109 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819776 - 0.734128i\)
\(L(\frac12)\) \(\approx\) \(0.819776 - 0.734128i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.402 + 1.68i)T \)
7 \( 1 + (-0.157 + 2.64i)T \)
good2 \( 1 + (-0.0159 - 0.00280i)T + (1.87 + 0.684i)T^{2} \)
5 \( 1 + (-3.75 - 1.36i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (1.76 + 4.84i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (0.220 - 0.606i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.69 + 2.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.328 + 0.189i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.993 - 0.175i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.97 - 5.41i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (2.20 - 6.06i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 - 6.16T + 37T^{2} \)
41 \( 1 + (-0.266 - 0.0969i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.45 - 8.22i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-6.91 + 2.51i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-1.71 + 0.992i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.62 - 2.20i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.75 + 7.57i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.63 - 9.29i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-8.57 + 4.94i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.08iT - 73T^{2} \)
79 \( 1 + (0.222 - 1.26i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (8.07 - 2.93i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (2.67 + 4.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.39 + 1.30i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.80140143075419997159395623802, −11.10173306295755237287431233188, −10.43037153223938641515371769865, −9.431974436269578145191131820065, −8.302638149423456721018087390275, −6.98034922645417734797632387048, −5.96533956360794596200441718414, −5.18543226776160894802675501363, −3.01805418033197642339351454531, −1.16496615970913042726666512638, 2.38884878189798592950316704726, 4.35862928247509445702655408326, 5.30564693104672539223613299969, 5.94686951288342220573150704820, 8.116380671761360686388880360893, 9.206881308836855062576847911115, 9.654715627262385585513047628787, 10.37025421764173889659585917555, 12.12053532604212563468919005072, 12.75444254577986543582946911651

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