Properties

Label 2-259182-1.1-c1-0-137
Degree $2$
Conductor $259182$
Sign $1$
Analytic cond. $2069.57$
Root an. cond. $45.4926$
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 + 4-s − 3·5-s − 7-s − 8-s + 3·10-s − 2·13-s + 14-s + 16-s + 17-s − 2·19-s − 3·20-s − 6·23-s + 4·25-s + 2·26-s − 28-s − 7·31-s − 32-s − 34-s + 3·35-s − 4·37-s + 2·38-s + 3·40-s + 6·41-s − 5·43-s + 6·46-s − 6·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.377·7-s − 0.353·8-s + 0.948·10-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.458·19-s − 0.670·20-s − 1.25·23-s + 4/5·25-s + 0.392·26-s − 0.188·28-s − 1.25·31-s − 0.176·32-s − 0.171·34-s + 0.507·35-s − 0.657·37-s + 0.324·38-s + 0.474·40-s + 0.937·41-s − 0.762·43-s + 0.884·46-s − 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(259182\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(2069.57\)
Root analytic conductor: \(45.4926\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{259182} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 259182,\ (\ :1/2),\ 1)\)

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 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−13.06225559816072, −12.76584426071708, −12.16330368628685, −11.74528839878980, −11.65028139223375, −10.87484103889441, −10.51400887516456, −10.09710825284227, −9.517444598928444, −9.103999689936403, −8.567068907346107, −8.091897142795288, −7.666201787705140, −7.424815150257543, −6.642410667749709, −6.512966105888521, −5.600323819638308, −5.284843143235936, −4.418194973265892, −4.057052742085610, −3.494794627346714, −3.063210709411415, −2.267979720676908, −1.799809868975243, −0.9289208537718159, 0, 0, 0.9289208537718159, 1.799809868975243, 2.267979720676908, 3.063210709411415, 3.494794627346714, 4.057052742085610, 4.418194973265892, 5.284843143235936, 5.600323819638308, 6.512966105888521, 6.642410667749709, 7.424815150257543, 7.666201787705140, 8.091897142795288, 8.567068907346107, 9.103999689936403, 9.517444598928444, 10.09710825284227, 10.51400887516456, 10.87484103889441, 11.65028139223375, 11.74528839878980, 12.16330368628685, 12.76584426071708, 13.06225559816072

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