Properties

Label 2-189618-1.1-c1-0-6
Degree $2$
Conductor $189618$
Sign $1$
Analytic cond. $1514.10$
Root an. cond. $38.9115$
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-s − 3-s + 4-s − 2·5-s + 6-s + 2·7-s − 8-s + 9-s + 2·10-s − 11-s − 12-s − 2·14-s + 2·15-s + 16-s + 17-s − 18-s + 2·19-s − 2·20-s − 2·21-s + 22-s + 2·23-s + 24-s − 25-s − 27-s + 2·28-s + 6·29-s − 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s − 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.458·19-s − 0.447·20-s − 0.436·21-s + 0.213·22-s + 0.417·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.377·28-s + 1.11·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189618\)    =    \(2 \cdot 3 \cdot 11 \cdot 13^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1514.10\)
Root analytic conductor: \(38.9115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{189618} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9434379983\)
\(L(\frac12)\) \(\approx\) \(0.9434379983\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.02265148289945, −12.48239803033873, −11.83691506180972, −11.77222740326303, −11.21900499511332, −10.82592572784817, −10.36666988915484, −9.838578809748302, −9.397949676647478, −8.585083643144185, −8.444101859424280, −7.795802759120839, −7.477237598908953, −6.907700213281318, −6.581440362057185, −5.636999626973970, −5.393586961076699, −4.851685110921111, −4.133160051527904, −3.729384670778755, −3.025109590558443, −2.347388051491151, −1.647926410212055, −1.022621178615562, −0.3785373028103345, 0.3785373028103345, 1.022621178615562, 1.647926410212055, 2.347388051491151, 3.025109590558443, 3.729384670778755, 4.133160051527904, 4.851685110921111, 5.393586961076699, 5.636999626973970, 6.581440362057185, 6.907700213281318, 7.477237598908953, 7.795802759120839, 8.444101859424280, 8.585083643144185, 9.397949676647478, 9.838578809748302, 10.36666988915484, 10.82592572784817, 11.21900499511332, 11.77222740326303, 11.83691506180972, 12.48239803033873, 13.02265148289945

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