Properties

Label 2-92400-1.1-c1-0-181
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 + 6·17-s − 3·19-s − 21-s + 5·23-s − 27-s − 7·29-s + 3·31-s + 33-s − 4·37-s − 3·39-s − 8·41-s + 13·43-s + 49-s − 6·51-s + 12·53-s + 3·57-s + 8·59-s − 10·61-s + 63-s + 12·67-s − 5·69-s − 5·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 1.45·17-s − 0.688·19-s − 0.218·21-s + 1.04·23-s − 0.192·27-s − 1.29·29-s + 0.538·31-s + 0.174·33-s − 0.657·37-s − 0.480·39-s − 1.24·41-s + 1.98·43-s + 1/7·49-s − 0.840·51-s + 1.64·53-s + 0.397·57-s + 1.04·59-s − 1.28·61-s + 0.125·63-s + 1.46·67-s − 0.601·69-s − 0.593·71-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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−13.93568331974353, −13.61110691334828, −13.09897919487133, −12.43477247857437, −12.26259793393521, −11.55359247018623, −11.09684985862575, −10.68408895265378, −10.29591319220165, −9.636763852439332, −9.136296019039590, −8.513872464180710, −8.079387121702900, −7.497210954418457, −6.968894208534039, −6.474917875934511, −5.677387940960073, −5.465908939493901, −4.960302812236064, −4.016742064695732, −3.829175582867647, −2.958632938753029, −2.312413289206777, −1.403071782354198, −1.003111811896128, 0, 1.003111811896128, 1.403071782354198, 2.312413289206777, 2.958632938753029, 3.829175582867647, 4.016742064695732, 4.960302812236064, 5.465908939493901, 5.677387940960073, 6.474917875934511, 6.968894208534039, 7.497210954418457, 8.079387121702900, 8.513872464180710, 9.136296019039590, 9.636763852439332, 10.29591319220165, 10.68408895265378, 11.09684985862575, 11.55359247018623, 12.26259793393521, 12.43477247857437, 13.09897919487133, 13.61110691334828, 13.93568331974353

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