Properties

Label 2-189618-1.1-c1-0-38
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s + 11-s − 12-s − 2·14-s + 16-s − 17-s − 18-s − 5·19-s − 2·21-s − 22-s − 9·23-s + 24-s − 5·25-s − 27-s + 2·28-s − 5·29-s + 8·31-s − 32-s − 33-s + 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.14·19-s − 0.436·21-s − 0.213·22-s − 1.87·23-s + 0.204·24-s − 25-s − 0.192·27-s + 0.377·28-s − 0.928·29-s + 1.43·31-s − 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-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: \(2\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.44423188495784, −13.09756756318677, −12.52513737047938, −11.97992003458327, −11.54028967126780, −11.29198789645661, −10.87935883442595, −10.10048577687131, −9.894983520726100, −9.520547690221538, −8.724413491155361, −8.210756156999241, −8.002510040968880, −7.516172375352143, −6.720811557681328, −6.382405989236890, −5.935709023789372, −5.444320145400147, −4.656132328334933, −4.212738215371121, −3.862756906761278, −2.846705324930367, −2.293704001059260, −1.605435372342959, −1.312224820501092, 0, 0, 1.312224820501092, 1.605435372342959, 2.293704001059260, 2.846705324930367, 3.862756906761278, 4.212738215371121, 4.656132328334933, 5.444320145400147, 5.935709023789372, 6.382405989236890, 6.720811557681328, 7.516172375352143, 8.002510040968880, 8.210756156999241, 8.724413491155361, 9.520547690221538, 9.894983520726100, 10.10048577687131, 10.87935883442595, 11.29198789645661, 11.54028967126780, 11.97992003458327, 12.52513737047938, 13.09756756318677, 13.44423188495784

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