Properties

Label 2-92400-1.1-c1-0-135
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 − 5·13-s − 3·17-s + 7·19-s − 21-s + 8·23-s + 27-s − 6·29-s − 33-s − 3·37-s − 5·39-s − 3·41-s + 4·43-s + 4·47-s + 49-s − 3·51-s − 7·53-s + 7·57-s − 2·59-s + 61-s − 63-s − 3·67-s + 8·69-s − 15·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s − 0.727·17-s + 1.60·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s − 1.11·29-s − 0.174·33-s − 0.493·37-s − 0.800·39-s − 0.468·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.420·51-s − 0.961·53-s + 0.927·57-s − 0.260·59-s + 0.128·61-s − 0.125·63-s − 0.366·67-s + 0.963·69-s − 1.78·71-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: $\chi_{92400} (1, \cdot )$
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 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.01100280179393, −13.56517186018102, −13.18051044562013, −12.69966971858000, −12.09637318931005, −11.80646482908370, −10.93361178809581, −10.75027965877984, −9.970635939076606, −9.460373035722815, −9.245356689762821, −8.739192244946252, −7.875019749751109, −7.579031971704078, −7.016217364138423, −6.736991586313536, −5.768685208967516, −5.284489835186899, −4.799389316888647, −4.223168122455577, −3.332269977971629, −3.052963223243075, −2.415541824282572, −1.751729543503856, −0.8840185621000157, 0, 0.8840185621000157, 1.751729543503856, 2.415541824282572, 3.052963223243075, 3.332269977971629, 4.223168122455577, 4.799389316888647, 5.284489835186899, 5.768685208967516, 6.736991586313536, 7.016217364138423, 7.579031971704078, 7.875019749751109, 8.739192244946252, 9.245356689762821, 9.460373035722815, 9.970635939076606, 10.75027965877984, 10.93361178809581, 11.80646482908370, 12.09637318931005, 12.69966971858000, 13.18051044562013, 13.56517186018102, 14.01100280179393

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