Properties

Label 2-189-1.1-c1-0-2
Degree $2$
Conductor $189$
Sign $1$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·5-s + 7-s + 6·11-s − 4·13-s + 4·16-s + 3·17-s + 2·19-s − 6·20-s − 6·23-s + 4·25-s − 2·28-s − 6·29-s − 4·31-s + 3·35-s − 7·37-s − 3·41-s − 43-s − 12·44-s + 9·47-s + 49-s + 8·52-s − 6·53-s + 18·55-s + 9·59-s − 10·61-s − 8·64-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s + 0.377·7-s + 1.80·11-s − 1.10·13-s + 16-s + 0.727·17-s + 0.458·19-s − 1.34·20-s − 1.25·23-s + 4/5·25-s − 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.507·35-s − 1.15·37-s − 0.468·41-s − 0.152·43-s − 1.80·44-s + 1.31·47-s + 1/7·49-s + 1.10·52-s − 0.824·53-s + 2.42·55-s + 1.17·59-s − 1.28·61-s − 64-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: yes
Arithmetic: yes
Character: $\chi_{189} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.245344016\)
\(L(\frac12)\) \(\approx\) \(1.245344016\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.58433731583081615557168762807, −11.80712467067776826965075311823, −10.22213033820361484191010028356, −9.540146555327995507333228878724, −8.905165354580512836683090859524, −7.46477788862935127887709071190, −6.04902466517769458594559190180, −5.13832633779419862585354738180, −3.79566092998735399945677628017, −1.69414023663856774565794959548, 1.69414023663856774565794959548, 3.79566092998735399945677628017, 5.13832633779419862585354738180, 6.04902466517769458594559190180, 7.46477788862935127887709071190, 8.905165354580512836683090859524, 9.540146555327995507333228878724, 10.22213033820361484191010028356, 11.80712467067776826965075311823, 12.58433731583081615557168762807

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