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Properties

Label 2-189-3.2-c2-0-12
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  − 0.722i·2-s + 3.47·4-s − 7.94i·5-s + 2.64·7-s − 5.40i·8-s − 5.74·10-s + 9.25i·11-s − 22.1·13-s − 1.91i·14-s + 10.0·16-s − 27.2i·17-s + 31.8·19-s − 27.6i·20-s + 6.68·22-s + 13.5i·23-s + ⋯
L(s)  = 1  − 0.361i·2-s + 0.869·4-s − 1.58i·5-s + 0.377·7-s − 0.675i·8-s − 0.574·10-s + 0.841i·11-s − 1.70·13-s − 0.136i·14-s + 0.625·16-s − 1.60i·17-s + 1.67·19-s − 1.38i·20-s + 0.303·22-s + 0.588i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.27140 - 1.27140i\)
\(L(\frac12)\) \(\approx\) \(1.27140 - 1.27140i\)
\(L(2)\) 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 \( 1 + 0.722iT - 4T^{2} \)
5 \( 1 + 7.94iT - 25T^{2} \)
11 \( 1 - 9.25iT - 121T^{2} \)
13 \( 1 + 22.1T + 169T^{2} \)
17 \( 1 + 27.2iT - 289T^{2} \)
19 \( 1 - 31.8T + 361T^{2} \)
23 \( 1 - 13.5iT - 529T^{2} \)
29 \( 1 + 6.73iT - 841T^{2} \)
31 \( 1 - 11.9T + 961T^{2} \)
37 \( 1 - 26.8T + 1.36e3T^{2} \)
41 \( 1 - 63.9iT - 1.68e3T^{2} \)
43 \( 1 + 16.2T + 1.84e3T^{2} \)
47 \( 1 - 0.518iT - 2.20e3T^{2} \)
53 \( 1 - 71.8iT - 2.80e3T^{2} \)
59 \( 1 - 38.5iT - 3.48e3T^{2} \)
61 \( 1 - 27.9T + 3.72e3T^{2} \)
67 \( 1 - 29.5T + 4.48e3T^{2} \)
71 \( 1 - 39.4iT - 5.04e3T^{2} \)
73 \( 1 - 67.1T + 5.32e3T^{2} \)
79 \( 1 + 18.7T + 6.24e3T^{2} \)
83 \( 1 - 80.0iT - 6.88e3T^{2} \)
89 \( 1 + 62.9iT - 7.92e3T^{2} \)
97 \( 1 + 46.7T + 9.40e3T^{2} \)
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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

−11.96113079715228265520056808970, −11.56831763677089310601634412029, −9.755463062016051583295097726129, −9.527098965185142540390871717709, −7.82518396594787692474883933165, −7.18701076991642243079408407702, −5.35700457484492641854227630263, −4.62510932171510529676916049330, −2.63859987119320154898442511320, −1.12814751984459438724577775041, 2.28398247401884222969156153547, 3.37459480461844376771359875881, 5.42540274591392708659735991127, 6.50826297022382866825580297001, 7.31695713159108757195373472276, 8.142812062235038470983888240953, 9.915012208670637069363109787213, 10.70517841607096667671420666049, 11.44643151638168686401382278405, 12.32876447158557714889027752047

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