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Properties

Label 2-92400-1.1-c1-0-103
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s + 11-s + 4·13-s + 5·17-s − 19-s − 21-s − 5·23-s + 27-s + 3·29-s + 6·31-s + 33-s + 12·37-s + 4·39-s − 2·41-s + 13·43-s − 6·47-s + 49-s + 5·51-s − 53-s − 57-s + 11·59-s + 5·61-s − 63-s − 10·67-s − 5·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 1.21·17-s − 0.229·19-s − 0.218·21-s − 1.04·23-s + 0.192·27-s + 0.557·29-s + 1.07·31-s + 0.174·33-s + 1.97·37-s + 0.640·39-s − 0.312·41-s + 1.98·43-s − 0.875·47-s + 1/7·49-s + 0.700·51-s − 0.137·53-s − 0.132·57-s + 1.43·59-s + 0.640·61-s − 0.125·63-s − 1.22·67-s − 0.601·69-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: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.422608020\)
\(L(\frac12)\) \(\approx\) \(4.422608020\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−13.90842411290709, −13.32259809811880, −12.97476338413649, −12.41921160803051, −11.85790338903620, −11.49402213905013, −10.81229446344107, −10.19412657465798, −9.962916611746052, −9.281904690423313, −8.922652816567435, −8.174042938402249, −7.950319815383253, −7.425108962975954, −6.584062915119228, −6.177773750117860, −5.851307968788708, −4.975435053192478, −4.356098329394070, −3.753844505395157, −3.410298944249394, −2.622585284349294, −2.125432427425057, −1.125839043632162, −0.7566147883433241, 0.7566147883433241, 1.125839043632162, 2.125432427425057, 2.622585284349294, 3.410298944249394, 3.753844505395157, 4.356098329394070, 4.975435053192478, 5.851307968788708, 6.177773750117860, 6.584062915119228, 7.425108962975954, 7.950319815383253, 8.174042938402249, 8.922652816567435, 9.281904690423313, 9.962916611746052, 10.19412657465798, 10.81229446344107, 11.49402213905013, 11.85790338903620, 12.41921160803051, 12.97476338413649, 13.32259809811880, 13.90842411290709

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