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Properties

Label 2-189-63.41-c1-0-4
Degree $2$
Conductor $189$
Sign $0.569 + 0.822i$
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  + (1.97 − 1.13i)2-s + (1.59 − 2.75i)4-s + (−0.717 + 1.24i)5-s + (2.16 − 1.52i)7-s − 2.69i·8-s + 3.26i·10-s + (−2.80 + 1.61i)11-s + (−4.43 − 2.55i)13-s + (2.52 − 5.46i)14-s + (0.119 + 0.207i)16-s + 1.09·17-s + 4.48i·19-s + (2.28 + 3.95i)20-s + (−3.68 + 6.37i)22-s + (−3.47 − 2.00i)23-s + ⋯
L(s)  = 1  + (1.39 − 0.804i)2-s + (0.795 − 1.37i)4-s + (−0.320 + 0.555i)5-s + (0.817 − 0.576i)7-s − 0.951i·8-s + 1.03i·10-s + (−0.844 + 0.487i)11-s + (−1.22 − 0.709i)13-s + (0.675 − 1.46i)14-s + (0.0298 + 0.0517i)16-s + 0.264·17-s + 1.02i·19-s + (0.510 + 0.883i)20-s + (−0.784 + 1.35i)22-s + (−0.723 − 0.417i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.569 + 0.822i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.569 + 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97704 - 1.03564i\)
\(L(\frac12)\) \(\approx\) \(1.97704 - 1.03564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.16 + 1.52i)T \)
good2 \( 1 + (-1.97 + 1.13i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (0.717 - 1.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.80 - 1.61i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.43 + 2.55i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.09T + 17T^{2} \)
19 \( 1 - 4.48iT - 19T^{2} \)
23 \( 1 + (3.47 + 2.00i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.02 - 0.593i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.24 - 1.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.239T + 37T^{2} \)
41 \( 1 + (-3.71 + 6.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.82 + 6.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.11 + 3.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.01iT - 53T^{2} \)
59 \( 1 + (4.73 - 8.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.82 + 1.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.330 - 0.571i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.82iT - 71T^{2} \)
73 \( 1 - 7.31iT - 73T^{2} \)
79 \( 1 + (1.83 + 3.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.45 + 9.44i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + (2.69 - 1.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.37020126852564543296614955014, −11.71949585910116902473558136567, −10.52223317160537809516397193544, −10.22780628863034258053843949314, −8.083451446390346511575896714812, −7.18774944894088821947460012875, −5.54867858174274211105772475266, −4.69099158626331895663462462267, −3.48679074714212651949736984927, −2.17910684348331092153140832136, 2.70897374221943998806039585883, 4.49587522070343263889297810480, 5.02641159710826919225303779519, 6.16096397987087651524095263886, 7.49270320227827133549069104588, 8.250795887119348265482098227398, 9.618132102576302245054999150946, 11.28198629767966383160236363855, 12.05929674419499887300246292951, 12.82044507812302554633952976846

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