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Properties

Label 2-189-21.5-c1-0-1
Degree $2$
Conductor $189$
Sign $0.444 - 0.895i$
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  + (−1.93 − 1.11i)2-s + (1.5 + 2.59i)4-s + (−2 + 1.73i)7-s − 2.23i·8-s + (−3.87 + 2.23i)11-s + 1.73i·13-s + (5.80 − 1.11i)14-s + (0.499 − 0.866i)16-s + (3.87 + 6.70i)17-s + (−3 − 1.73i)19-s + 10.0·22-s + (3.87 + 2.23i)23-s + (2.5 + 4.33i)25-s + (1.93 − 3.35i)26-s + (−7.50 − 2.59i)28-s − 4.47i·29-s + ⋯
L(s)  = 1  + (−1.36 − 0.790i)2-s + (0.750 + 1.29i)4-s + (−0.755 + 0.654i)7-s − 0.790i·8-s + (−1.16 + 0.674i)11-s + 0.480i·13-s + (1.55 − 0.298i)14-s + (0.124 − 0.216i)16-s + (0.939 + 1.62i)17-s + (−0.688 − 0.397i)19-s + 2.13·22-s + (0.807 + 0.466i)23-s + (0.5 + 0.866i)25-s + (0.379 − 0.657i)26-s + (−1.41 − 0.490i)28-s − 0.830i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.318655 + 0.197694i\)
\(L(\frac12)\) \(\approx\) \(0.318655 + 0.197694i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good2 \( 1 + (1.93 + 1.11i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.87 - 2.23i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + (-3.87 - 6.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.87 - 2.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.74T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (-3.87 + 6.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.87 - 2.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.87 + 6.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.5 - 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.94iT - 71T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.74T + 83T^{2} \)
89 \( 1 + (-7.74 + 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.73iT - 97T^{2} \)
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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

−12.53387953061731117403519064093, −11.56119830353043660097877924199, −10.46109905123792561713035439922, −9.904251074649682331447935489016, −8.877375778088524768538455432004, −8.070332322474198534288691873495, −6.85556458445052996666685167834, −5.32978516995454780332911919535, −3.27985420042283977712604627975, −1.94501038276519169271580926239, 0.50477077918082115811405497219, 3.13133603822800027139059085270, 5.24364929224346096197510818234, 6.52165438393630029415990490241, 7.42042119261756214635580008847, 8.272125857405220080718842926905, 9.307977702143290251685248366816, 10.28939772636855743512463815801, 10.79039914182298604798030243901, 12.44903120588937644615478274898

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