Properties

Label 2-72450-1.1-c1-0-130
Degree $2$
Conductor $72450$
Sign $1$
Analytic cond. $578.516$
Root an. cond. $24.0523$
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 − 7-s − 8-s − 2·11-s − 2·13-s + 14-s + 16-s + 2·17-s − 4·19-s + 2·22-s − 23-s + 2·26-s − 28-s + 2·29-s − 8·31-s − 32-s − 2·34-s − 4·37-s + 4·38-s − 12·41-s − 2·43-s − 2·44-s + 46-s − 8·47-s + 49-s − 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.603·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.426·22-s − 0.208·23-s + 0.392·26-s − 0.188·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 0.657·37-s + 0.648·38-s − 1.87·41-s − 0.304·43-s − 0.301·44-s + 0.147·46-s − 1.16·47-s + 1/7·49-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(578.516\)
Root analytic conductor: \(24.0523\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{72450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.61302344246187, −14.25772325014356, −13.51659926213193, −13.01721398881188, −12.54027842735706, −12.17253390536359, −11.45919921539619, −11.07067357997010, −10.41988992261632, −10.01460652416003, −9.710919110710429, −8.961609420472137, −8.493827577891492, −8.060277912494638, −7.447465849167627, −6.916100594080059, −6.495990353083944, −5.792639631916627, −5.219442342887051, −4.712500099085165, −3.813672838397534, −3.277571528518989, −2.645484584937226, −1.924696681635766, −1.368409995642201, 0, 0, 1.368409995642201, 1.924696681635766, 2.645484584937226, 3.277571528518989, 3.813672838397534, 4.712500099085165, 5.219442342887051, 5.792639631916627, 6.495990353083944, 6.916100594080059, 7.447465849167627, 8.060277912494638, 8.493827577891492, 8.961609420472137, 9.710919110710429, 10.01460652416003, 10.41988992261632, 11.07067357997010, 11.45919921539619, 12.17253390536359, 12.54027842735706, 13.01721398881188, 13.51659926213193, 14.25772325014356, 14.61302344246187

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