Properties

Label 2-189618-1.1-c1-0-20
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 + 3·7-s + 8-s + 9-s + 2·10-s − 11-s − 12-s + 3·14-s − 2·15-s + 16-s − 17-s + 18-s + 4·19-s + 2·20-s − 3·21-s − 22-s + 2·23-s − 24-s − 25-s − 27-s + 3·28-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 + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s + 0.801·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.654·21-s − 0.213·22-s + 0.417·23-s − 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.566·28-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

\(L(1)\) \(\approx\) \(5.563896126\)
\(L(\frac12)\) \(\approx\) \(5.563896126\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.17087296319707, −12.69205835366440, −12.11911515415670, −11.74690623153345, −11.28499790356878, −10.82543511621843, −10.49115560006410, −9.918711372677426, −9.303144286735248, −9.038355751227564, −8.029030495973424, −7.923448143187580, −7.179303247011806, −6.818973979733770, −6.085756092882981, −5.733690893679973, −5.211647606178845, −4.984409931231206, −4.306983734903789, −3.798931340216131, −3.039965511177985, −2.391825828363100, −1.858387825281029, −1.353976658662409, −0.6131270377437798, 0.6131270377437798, 1.353976658662409, 1.858387825281029, 2.391825828363100, 3.039965511177985, 3.798931340216131, 4.306983734903789, 4.984409931231206, 5.211647606178845, 5.733690893679973, 6.085756092882981, 6.818973979733770, 7.179303247011806, 7.923448143187580, 8.029030495973424, 9.038355751227564, 9.303144286735248, 9.918711372677426, 10.49115560006410, 10.82543511621843, 11.28499790356878, 11.74690623153345, 12.11911515415670, 12.69205835366440, 13.17087296319707

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