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Properties

Label 2-189-21.20-c1-0-8
Degree $2$
Conductor $189$
Sign $-0.654 + 0.755i$
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  − 2.23i·2-s − 3.00·4-s + 3.87·5-s + (−2 − 1.73i)7-s + 2.23i·8-s − 8.66i·10-s − 2.23i·11-s + 3.46i·13-s + (−3.87 + 4.47i)14-s − 0.999·16-s + 5.19i·19-s − 11.6·20-s − 5.00·22-s − 2.23i·23-s + 10.0·25-s + 7.74·26-s + ⋯
L(s)  = 1  − 1.58i·2-s − 1.50·4-s + 1.73·5-s + (−0.755 − 0.654i)7-s + 0.790i·8-s − 2.73i·10-s − 0.674i·11-s + 0.960i·13-s + (−1.03 + 1.19i)14-s − 0.249·16-s + 1.19i·19-s − 2.59·20-s − 1.06·22-s − 0.466i·23-s + 2.00·25-s + 1.51·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.654 + 0.755i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.547839 - 1.19916i\)
\(L(\frac12)\) \(\approx\) \(0.547839 - 1.19916i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT - 2T^{2} \)
5 \( 1 - 3.87T + 5T^{2} \)
11 \( 1 + 2.23iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + 2.23iT - 23T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 3.87T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 7.74T + 47T^{2} \)
53 \( 1 + 8.94iT - 53T^{2} \)
59 \( 1 + 7.74T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 - 11.1iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 - 7.74T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−12.29820213580494058706517612021, −11.00897137844238312897365585340, −10.27248587092429499816909287478, −9.629390018444784495463203215370, −8.836656498878657218387275522995, −6.78709535216385006703237089919, −5.72926167363061150772777409765, −4.09457980771662169030647271265, −2.76691047005576429478913120071, −1.46341883486654928671535527742, 2.54379477291757560647864572511, 4.95738757290599019952705139486, 5.82526903388341409099158626797, 6.46441248412202969167806553322, 7.58381904159027412402719836443, 9.041912193689631135262666294743, 9.457769710290539094557814215942, 10.58231632012419085804117972913, 12.41542773185722704789239987634, 13.35078624594296651534536574375

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