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Properties

Label 2-72450-1.1-c1-0-78
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 − 3·11-s + 4·13-s + 14-s + 16-s + 6·17-s − 19-s + 3·22-s − 23-s − 4·26-s − 28-s + 2·31-s − 32-s − 6·34-s − 2·37-s + 38-s + 9·41-s − 2·43-s − 3·44-s + 46-s − 9·47-s + 49-s + 4·52-s + 9·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.904·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.229·19-s + 0.639·22-s − 0.208·23-s − 0.784·26-s − 0.188·28-s + 0.359·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s + 0.162·38-s + 1.40·41-s − 0.304·43-s − 0.452·44-s + 0.147·46-s − 1.31·47-s + 1/7·49-s + 0.554·52-s + 1.23·53-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: Trivial
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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−14.34471345238204, −13.90831969479551, −13.15683535299516, −12.95594016499945, −12.32700277149136, −11.74378201276452, −11.31135879019243, −10.70057268075000, −10.25893811107299, −9.870370541342910, −9.377022300753873, −8.556383541749081, −8.375065698479185, −7.760195008858453, −7.268784481665229, −6.642751893023041, −6.057996765448835, −5.565392979305619, −5.094906871858088, −4.082597050140318, −3.637034728076307, −2.902713317409872, −2.432073614286025, −1.473747040320245, −0.9171308682995363, 0, 0.9171308682995363, 1.473747040320245, 2.432073614286025, 2.902713317409872, 3.637034728076307, 4.082597050140318, 5.094906871858088, 5.565392979305619, 6.057996765448835, 6.642751893023041, 7.268784481665229, 7.760195008858453, 8.375065698479185, 8.556383541749081, 9.377022300753873, 9.870370541342910, 10.25893811107299, 10.70057268075000, 11.31135879019243, 11.74378201276452, 12.32700277149136, 12.95594016499945, 13.15683535299516, 13.90831969479551, 14.34471345238204

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