Properties

Label 2-72450-1.1-c1-0-92
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 − 2·13-s − 14-s + 16-s + 6·17-s − 4·19-s − 23-s − 2·26-s − 28-s − 6·29-s − 4·31-s + 32-s + 6·34-s − 2·37-s − 4·38-s + 6·41-s − 8·43-s − 46-s + 12·47-s + 49-s − 2·52-s − 6·53-s − 56-s − 6·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.208·23-s − 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.328·37-s − 0.648·38-s + 0.937·41-s − 1.21·43-s − 0.147·46-s + 1.75·47-s + 1/7·49-s − 0.277·52-s − 0.824·53-s − 0.133·56-s − 0.787·58-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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.45505596691552, −13.87261677947151, −13.25144952142663, −12.91056377612737, −12.35984149050036, −12.02239329290398, −11.50624386460036, −10.71783101193543, −10.54381012043473, −9.777967111809772, −9.446831514174184, −8.744945739305967, −8.126769316493535, −7.487739921455522, −7.214474763558102, −6.481395570013934, −5.923333862041864, −5.470181369260840, −4.959720877379336, −4.193558580000975, −3.693916990337529, −3.200292801849616, −2.406490460158183, −1.899584507198845, −0.9826577168300783, 0, 0.9826577168300783, 1.899584507198845, 2.406490460158183, 3.200292801849616, 3.693916990337529, 4.193558580000975, 4.959720877379336, 5.470181369260840, 5.923333862041864, 6.481395570013934, 7.214474763558102, 7.487739921455522, 8.126769316493535, 8.744945739305967, 9.446831514174184, 9.777967111809772, 10.54381012043473, 10.71783101193543, 11.50624386460036, 12.02239329290398, 12.35984149050036, 12.91056377612737, 13.25144952142663, 13.87261677947151, 14.45505596691552

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