Properties

Label 2-189-3.2-c2-0-11
Degree $2$
Conductor $189$
Sign $i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.928i·2-s + 3.13·4-s − 4.06i·5-s − 2.64·7-s − 6.62i·8-s − 3.77·10-s − 9.90i·11-s + 9.26·13-s + 2.45i·14-s + 6.40·16-s + 7.63i·17-s − 13.1·19-s − 12.7i·20-s − 9.19·22-s − 33.2i·23-s + ⋯
L(s)  = 1  − 0.464i·2-s + 0.784·4-s − 0.812i·5-s − 0.377·7-s − 0.828i·8-s − 0.377·10-s − 0.900i·11-s + 0.713·13-s + 0.175i·14-s + 0.400·16-s + 0.449i·17-s − 0.690·19-s − 0.637i·20-s − 0.417·22-s − 1.44i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.22275 - 1.22275i\)
\(L(\frac12)\) \(\approx\) \(1.22275 - 1.22275i\)
\(L(2)\) 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 \( 1 + 2.64T \)
good2 \( 1 + 0.928iT - 4T^{2} \)
5 \( 1 + 4.06iT - 25T^{2} \)
11 \( 1 + 9.90iT - 121T^{2} \)
13 \( 1 - 9.26T + 169T^{2} \)
17 \( 1 - 7.63iT - 289T^{2} \)
19 \( 1 + 13.1T + 361T^{2} \)
23 \( 1 + 33.2iT - 529T^{2} \)
29 \( 1 - 12.8iT - 841T^{2} \)
31 \( 1 - 22.1T + 961T^{2} \)
37 \( 1 + 56.7T + 1.36e3T^{2} \)
41 \( 1 - 31.1iT - 1.68e3T^{2} \)
43 \( 1 - 55.1T + 1.84e3T^{2} \)
47 \( 1 - 56.4iT - 2.20e3T^{2} \)
53 \( 1 - 83.3iT - 2.80e3T^{2} \)
59 \( 1 - 112. iT - 3.48e3T^{2} \)
61 \( 1 - 39.7T + 3.72e3T^{2} \)
67 \( 1 - 93.9T + 4.48e3T^{2} \)
71 \( 1 - 11.1iT - 5.04e3T^{2} \)
73 \( 1 + 39.2T + 5.32e3T^{2} \)
79 \( 1 - 44.3T + 6.24e3T^{2} \)
83 \( 1 + 118. iT - 6.88e3T^{2} \)
89 \( 1 - 110. iT - 7.92e3T^{2} \)
97 \( 1 + 138.T + 9.40e3T^{2} \)
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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

−12.24137239691939695991816284391, −10.98830898228759053425686360861, −10.44339057998220889566277296698, −9.025470066333639664138847444979, −8.233584314592992775856148007216, −6.72012880682844456471029668710, −5.84999921228652272429517724302, −4.19192259538390413989128578290, −2.82224312678982177941555818626, −1.08255699745915902361352106185, 2.14664810008875027767639423945, 3.54329925873822736092938473837, 5.37546610529332607986084618053, 6.61170547756635686402607463262, 7.13442396247840795112385888768, 8.321442919246827842186050997019, 9.728992056154674149365663026210, 10.68898663611234763846979965973, 11.49888517218669100513756903391, 12.49351103959219522675999135276

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