Properties

Label 2-189-1.1-c1-0-0
Degree $2$
Conductor $189$
Sign $1$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s + 0.999·4-s − 1.73·5-s + 7-s + 1.73·8-s + 2.99·10-s + 1.73·11-s + 2·13-s − 1.73·14-s − 5·16-s + 6.92·17-s + 5·19-s − 1.73·20-s − 2.99·22-s − 1.73·23-s − 2.00·25-s − 3.46·26-s + 0.999·28-s + 10.3·29-s + 5·31-s + 5.19·32-s − 11.9·34-s − 1.73·35-s − 7·37-s − 8.66·38-s − 3.00·40-s − 5.19·41-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.499·4-s − 0.774·5-s + 0.377·7-s + 0.612·8-s + 0.948·10-s + 0.522·11-s + 0.554·13-s − 0.462·14-s − 1.25·16-s + 1.68·17-s + 1.14·19-s − 0.387·20-s − 0.639·22-s − 0.361·23-s − 0.400·25-s − 0.679·26-s + 0.188·28-s + 1.92·29-s + 0.898·31-s + 0.918·32-s − 2.05·34-s − 0.292·35-s − 1.15·37-s − 1.40·38-s − 0.474·40-s − 0.811·41-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.6133065395\)
\(L(\frac12)\) \(\approx\) \(0.6133065395\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + 1.73T + 2T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + 1.73T + 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + 5.19T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 14T + 67T^{2} \)
71 \( 1 + 5.19T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 8.66T + 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−12.09640124844614307896764199575, −11.55842481276610500182624137908, −10.33215935458044833050157848405, −9.624661334042765996308778084156, −8.313039181832919097379508448372, −7.941762634148002301268447444519, −6.71299199100972000939995934606, −5.00760306973419028260098351078, −3.53880091685523403534063731882, −1.19573202810142059571855967422, 1.19573202810142059571855967422, 3.53880091685523403534063731882, 5.00760306973419028260098351078, 6.71299199100972000939995934606, 7.941762634148002301268447444519, 8.313039181832919097379508448372, 9.624661334042765996308778084156, 10.33215935458044833050157848405, 11.55842481276610500182624137908, 12.09640124844614307896764199575

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