Properties

Label 2-92400-1.1-c1-0-106
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 + 6·17-s + 21-s + 2·23-s + 27-s + 8·29-s + 10·31-s + 33-s − 8·37-s − 4·39-s + 10·41-s + 4·43-s + 49-s + 6·51-s + 10·53-s + 2·61-s + 63-s + 16·67-s + 2·69-s + 8·71-s + 12·73-s + 77-s + 10·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 1.45·17-s + 0.218·21-s + 0.417·23-s + 0.192·27-s + 1.48·29-s + 1.79·31-s + 0.174·33-s − 1.31·37-s − 0.640·39-s + 1.56·41-s + 0.609·43-s + 1/7·49-s + 0.840·51-s + 1.37·53-s + 0.256·61-s + 0.125·63-s + 1.95·67-s + 0.240·69-s + 0.949·71-s + 1.40·73-s + 0.113·77-s + 1.12·79-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: $\chi_{92400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.639307381\)
\(L(\frac12)\) \(\approx\) \(4.639307381\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.00000789028131, −13.54025124991266, −12.71125881726532, −12.24827472306455, −12.14042712499743, −11.44656800107859, −10.80130900964709, −10.22275613261496, −9.892677141849115, −9.419923157347278, −8.812883493089170, −8.186356142035107, −7.948262201113357, −7.323093618472946, −6.781640370511209, −6.324100061804298, −5.382960922106140, −5.146834731360652, −4.435884243390860, −3.899615564970871, −3.213248342568338, −2.582319907489914, −2.202114049508571, −1.111372715940850, −0.7808181934198037, 0.7808181934198037, 1.111372715940850, 2.202114049508571, 2.582319907489914, 3.213248342568338, 3.899615564970871, 4.435884243390860, 5.146834731360652, 5.382960922106140, 6.324100061804298, 6.781640370511209, 7.323093618472946, 7.948262201113357, 8.186356142035107, 8.812883493089170, 9.419923157347278, 9.892677141849115, 10.22275613261496, 10.80130900964709, 11.44656800107859, 12.14042712499743, 12.24827472306455, 12.71125881726532, 13.54025124991266, 14.00000789028131

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