Properties

Label 2-92400-1.1-c1-0-132
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 + 6·13-s + 7·17-s + 5·19-s + 21-s − 23-s + 27-s − 5·29-s + 8·31-s − 33-s + 2·37-s + 6·39-s + 12·41-s − 11·43-s + 8·47-s + 49-s + 7·51-s + 11·53-s + 5·57-s + 5·59-s + 7·61-s + 63-s − 2·67-s − 69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s + 1.69·17-s + 1.14·19-s + 0.218·21-s − 0.208·23-s + 0.192·27-s − 0.928·29-s + 1.43·31-s − 0.174·33-s + 0.328·37-s + 0.960·39-s + 1.87·41-s − 1.67·43-s + 1.16·47-s + 1/7·49-s + 0.980·51-s + 1.51·53-s + 0.662·57-s + 0.650·59-s + 0.896·61-s + 0.125·63-s − 0.244·67-s − 0.120·69-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: $\chi_{92400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.426892096\)
\(L(\frac12)\) \(\approx\) \(5.426892096\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−13.73104580849294, −13.47897523382686, −13.04374317941826, −12.33684933101140, −11.73270331354457, −11.59235897687794, −10.77303033129151, −10.34214960895786, −9.910817031291187, −9.277063864466414, −8.867052347214395, −8.150382626681459, −7.953935274395279, −7.419463369792980, −6.818189268701385, −6.035050550280048, −5.635277093367067, −5.194010802617369, −4.299317713920312, −3.833207329551018, −3.286813006982238, −2.765429970078676, −1.974451404971875, −1.117146724243740, −0.8780917576656054, 0.8780917576656054, 1.117146724243740, 1.974451404971875, 2.765429970078676, 3.286813006982238, 3.833207329551018, 4.299317713920312, 5.194010802617369, 5.635277093367067, 6.035050550280048, 6.818189268701385, 7.419463369792980, 7.953935274395279, 8.150382626681459, 8.867052347214395, 9.277063864466414, 9.910817031291187, 10.34214960895786, 10.77303033129151, 11.59235897687794, 11.73270331354457, 12.33684933101140, 13.04374317941826, 13.47897523382686, 13.73104580849294

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