Properties

Label 2-189-189.47-c1-0-5
Degree $2$
Conductor $189$
Sign $0.726 - 0.687i$
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  + (−2.68 − 0.472i)2-s + (0.605 + 1.62i)3-s + (5.08 + 1.85i)4-s + (2.55 + 0.930i)5-s + (−0.856 − 4.63i)6-s + (2.58 − 0.579i)7-s + (−8.05 − 4.65i)8-s + (−2.26 + 1.96i)9-s + (−6.41 − 3.70i)10-s + (0.0485 + 0.133i)11-s + (0.0748 + 9.38i)12-s + (0.695 − 1.91i)13-s + (−7.19 + 0.333i)14-s + (0.0376 + 4.71i)15-s + (11.1 + 9.32i)16-s + (2.00 − 3.46i)17-s + ⋯
L(s)  = 1  + (−1.89 − 0.334i)2-s + (0.349 + 0.936i)3-s + (2.54 + 0.926i)4-s + (1.14 + 0.416i)5-s + (−0.349 − 1.89i)6-s + (0.975 − 0.219i)7-s + (−2.84 − 1.64i)8-s + (−0.755 + 0.654i)9-s + (−2.02 − 1.17i)10-s + (0.0146 + 0.0401i)11-s + (0.0216 + 2.70i)12-s + (0.192 − 0.530i)13-s + (−1.92 + 0.0892i)14-s + (0.00971 + 1.21i)15-s + (2.77 + 2.33i)16-s + (0.485 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.726 - 0.687i$
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: \((2,\ 189,\ (\ :1/2),\ 0.726 - 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.691659 + 0.275474i\)
\(L(\frac12)\) \(\approx\) \(0.691659 + 0.275474i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.605 - 1.62i)T \)
7 \( 1 + (-2.58 + 0.579i)T \)
good2 \( 1 + (2.68 + 0.472i)T + (1.87 + 0.684i)T^{2} \)
5 \( 1 + (-2.55 - 0.930i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (-0.0485 - 0.133i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-0.695 + 1.91i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.00 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.63 - 1.52i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.29 + 0.228i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.12 - 5.84i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.45 - 3.99i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + 9.38T + 37T^{2} \)
41 \( 1 + (0.303 + 0.110i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.643 + 3.65i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (6.27 - 2.28i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-6.59 + 3.81i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.94 - 4.14i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (3.87 + 10.6i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.311 + 1.76i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-6.03 + 3.48i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.24iT - 73T^{2} \)
79 \( 1 + (2.60 - 14.7i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.966 + 0.351i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (2.58 + 4.47i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.25 - 1.27i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.18934950972806593976951291391, −10.94795088465647190310666598381, −10.53107502993900375219067864697, −9.769352222398963413853893661763, −8.865010896810209338798289477060, −8.082353949648952211717980745656, −6.87676606091905298977113319182, −5.38072074866577307891063434528, −3.13412988280081230442441654341, −1.83389905505896153872890540704, 1.41216488355902429212447881887, 2.23633673390053963615291230324, 5.67258438901894908176175182023, 6.50329463741744358649713986673, 7.66410439409195246075540244979, 8.525742659456632302533863014095, 9.112122003724016249454943959793, 10.16736629412343064436185454584, 11.24722216215990942839310089683, 12.14594509598000260163299760893

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