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Properties

Label 2-189-21.11-c0-0-0
Degree $2$
Conductor $189$
Sign $0.895 - 0.444i$
Analytic cond. $0.0943232$
Root an. cond. $0.307120$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)4-s + 7-s − 13-s + (−0.499 − 0.866i)16-s + (−1 − 1.73i)19-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s − 43-s + 49-s + (0.5 − 0.866i)52-s + (0.5 + 0.866i)61-s + 0.999·64-s + (0.5 − 0.866i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + 7-s − 13-s + (−0.499 − 0.866i)16-s + (−1 − 1.73i)19-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s − 43-s + 49-s + (0.5 − 0.866i)52-s + (0.5 + 0.866i)61-s + 0.999·64-s + (0.5 − 0.866i)67-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(0.0943232\)
Root analytic conductor: \(0.307120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :0),\ 0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6520774674\)
\(L(\frac12)\) \(\approx\) \(0.6520774674\)
\(L(1)\) 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 \( 1 - T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−12.95712025298803857390028926301, −11.84183935124613510867712083098, −11.18394374517780049218032169032, −9.771200136271633034970017040564, −8.752839438830055882330236675064, −7.88639050962153812660136432658, −6.92610513305897376346028673064, −5.11203554069154677044449361215, −4.26297223858260436199025984889, −2.55939948811483532521196998680, 1.91255076604218535700168097614, 4.20744014396295569953301800008, 5.19214167103579096857904269941, 6.30087174016502298916855520295, 7.80499230711910464163221183475, 8.703775877156780684159818138802, 9.962045875563006616284826861808, 10.56154615413713456608029130908, 11.76185879594565457888328419177, 12.69411025483152826804369717737

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