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Properties

Label 2-189-63.11-c2-0-7
Degree $2$
Conductor $189$
Sign $0.906 - 0.423i$
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  + 0.513i·2-s + 3.73·4-s + (6.08 + 3.51i)5-s + (1.23 − 6.88i)7-s + 3.97i·8-s + (−1.80 + 3.12i)10-s + (−3.15 + 1.82i)11-s + (−3.79 − 6.56i)13-s + (3.53 + 0.636i)14-s + 12.9·16-s + (−17.5 − 10.1i)17-s + (13.6 + 23.7i)19-s + (22.7 + 13.1i)20-s + (−0.937 − 1.62i)22-s + (−3.42 − 1.97i)23-s + ⋯
L(s)  = 1  + 0.256i·2-s + 0.934·4-s + (1.21 + 0.702i)5-s + (0.177 − 0.984i)7-s + 0.496i·8-s + (−0.180 + 0.312i)10-s + (−0.287 + 0.165i)11-s + (−0.291 − 0.505i)13-s + (0.252 + 0.0454i)14-s + 0.806·16-s + (−1.03 − 0.594i)17-s + (0.720 + 1.24i)19-s + (1.13 + 0.656i)20-s + (−0.0425 − 0.0737i)22-s + (−0.148 − 0.0858i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.906 - 0.423i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.906 - 0.423i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.11972 + 0.470641i\)
\(L(\frac12)\) \(\approx\) \(2.11972 + 0.470641i\)
\(L(2)\) 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 \( 1 + (-1.23 + 6.88i)T \)
good2 \( 1 - 0.513iT - 4T^{2} \)
5 \( 1 + (-6.08 - 3.51i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (3.15 - 1.82i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (3.79 + 6.56i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (17.5 + 10.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-13.6 - 23.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (3.42 + 1.97i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-23.7 - 13.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 4.85T + 961T^{2} \)
37 \( 1 + (18.7 + 32.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (61.1 - 35.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (9.41 - 16.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 23.9iT - 2.20e3T^{2} \)
53 \( 1 + (23.1 + 13.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + 52.8iT - 3.48e3T^{2} \)
61 \( 1 + 106.T + 3.72e3T^{2} \)
67 \( 1 - 102.T + 4.48e3T^{2} \)
71 \( 1 + 138. iT - 5.04e3T^{2} \)
73 \( 1 + (34.7 - 60.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 23.2T + 6.24e3T^{2} \)
83 \( 1 + (-25.9 - 14.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (135. - 78.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-2.93 + 5.07i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.40511752356884614740769770007, −11.18455514859787912147652878873, −10.40148376128037436183415061773, −9.795549111428560318661620490824, −8.066964330168864639552200716228, −7.05879676870957399529407473241, −6.32173534435729624707313570785, −5.12349813390535949389985584180, −3.14026846836568415154744004069, −1.82254770494106086080348245728, 1.71342291786928751993249200261, 2.71203781294922486480277356670, 4.89113680136319464758560062082, 5.92408844183070379237808070578, 6.86224600055972140772784711197, 8.457374534165638476591374602326, 9.317858469443288140983204271336, 10.28102849322232887922927757043, 11.41265833236450931142196842646, 12.18702175675026621517336940397

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