Properties

Label 2-72450-1.1-c1-0-51
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 $1$

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 − 8·29-s + 2·31-s − 32-s − 2·34-s − 4·37-s + 4·38-s − 2·41-s + 8·43-s − 2·44-s + 46-s + 12·47-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 − 1.48·29-s + 0.359·31-s − 0.176·32-s − 0.342·34-s − 0.657·37-s + 0.648·38-s − 0.312·41-s + 1.21·43-s − 0.301·44-s + 0.147·46-s + 1.75·47-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: \(1\)
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 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−14.40878170241767, −13.84347364156769, −13.32391847136367, −12.73072879926328, −12.33360668605962, −11.90353164335959, −11.21867402111800, −10.65089935233217, −10.37206563918271, −9.833420442524894, −9.190557926338249, −8.917298593967909, −8.208452597578762, −7.631818306986336, −7.356131355048365, −6.651208156650070, −6.139800542988928, −5.439756686922243, −5.127581738406975, −4.011910087446845, −3.815880309542261, −2.731041629323965, −2.434606927864632, −1.679596287637530, −0.7355730172935726, 0, 0.7355730172935726, 1.679596287637530, 2.434606927864632, 2.731041629323965, 3.815880309542261, 4.011910087446845, 5.127581738406975, 5.439756686922243, 6.139800542988928, 6.651208156650070, 7.356131355048365, 7.631818306986336, 8.208452597578762, 8.917298593967909, 9.190557926338249, 9.833420442524894, 10.37206563918271, 10.65089935233217, 11.21867402111800, 11.90353164335959, 12.33360668605962, 12.73072879926328, 13.32391847136367, 13.84347364156769, 14.40878170241767

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