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Properties

Label 2-92400-1.1-c1-0-152
Degree $2$
Conductor $92400$
Sign $-1$
Analytic cond. $737.817$
Root an. cond. $27.1628$
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  − 3-s + 7-s + 9-s − 11-s + 3·13-s + 3·17-s − 21-s + 5·23-s − 27-s − 7·29-s − 9·31-s + 33-s + 8·37-s − 3·39-s − 5·41-s + 43-s + 6·47-s + 49-s − 3·51-s + 3·53-s − 13·59-s − 7·61-s + 63-s − 12·67-s − 5·69-s − 8·71-s + 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.727·17-s − 0.218·21-s + 1.04·23-s − 0.192·27-s − 1.29·29-s − 1.61·31-s + 0.174·33-s + 1.31·37-s − 0.480·39-s − 0.780·41-s + 0.152·43-s + 0.875·47-s + 1/7·49-s − 0.420·51-s + 0.412·53-s − 1.69·59-s − 0.896·61-s + 0.125·63-s − 1.46·67-s − 0.601·69-s − 0.949·71-s + 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(737.817\)
Root analytic conductor: \(27.1628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−14.10622071859260, −13.38586025202891, −13.17913336540330, −12.58752317093076, −12.15734840755541, −11.40612302178803, −11.26327970674677, −10.62199999097374, −10.36708198785004, −9.580159692935436, −8.975619027472175, −8.834928537637287, −7.791941941623439, −7.579731251137447, −7.132327553533860, −6.223176144180867, −5.938364205498262, −5.388143392249597, −4.844597681730955, −4.272041202556212, −3.538593714171780, −3.141247480098576, −2.182003999137145, −1.535873180811284, −0.9214946961828238, 0, 0.9214946961828238, 1.535873180811284, 2.182003999137145, 3.141247480098576, 3.538593714171780, 4.272041202556212, 4.844597681730955, 5.388143392249597, 5.938364205498262, 6.223176144180867, 7.132327553533860, 7.579731251137447, 7.791941941623439, 8.834928537637287, 8.975619027472175, 9.580159692935436, 10.36708198785004, 10.62199999097374, 11.26327970674677, 11.40612302178803, 12.15734840755541, 12.58752317093076, 13.17913336540330, 13.38586025202891, 14.10622071859260

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