Properties

Label 2-92400-1.1-c1-0-68
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 + 13-s + 7·17-s − 3·19-s + 21-s + 27-s − 2·29-s + 4·31-s − 33-s − 3·37-s + 39-s − 9·41-s + 8·47-s + 49-s + 7·51-s + 9·53-s − 3·57-s + 10·59-s + 11·61-s + 63-s + 13·67-s + 71-s − 5·73-s − 77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 1.69·17-s − 0.688·19-s + 0.218·21-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.174·33-s − 0.493·37-s + 0.160·39-s − 1.40·41-s + 1.16·47-s + 1/7·49-s + 0.980·51-s + 1.23·53-s − 0.397·57-s + 1.30·59-s + 1.40·61-s + 0.125·63-s + 1.58·67-s + 0.118·71-s − 0.585·73-s − 0.113·77-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

\(L(1)\) \(\approx\) \(3.923841142\)
\(L(\frac12)\) \(\approx\) \(3.923841142\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−13.97237809901606, −13.32835732248895, −12.94945937579698, −12.40990392217732, −11.79028692896229, −11.58147146738998, −10.71683579347027, −10.26912410283752, −10.00804554251720, −9.353708207724643, −8.616070600930471, −8.440886089941148, −7.853549260518869, −7.350926442161510, −6.826358166142036, −6.201227035762985, −5.421400488859558, −5.230263512348078, −4.407632517289687, −3.695276382000266, −3.462913031987748, −2.534949156797541, −2.119076882318754, −1.271091342676802, −0.6470016406614035, 0.6470016406614035, 1.271091342676802, 2.119076882318754, 2.534949156797541, 3.462913031987748, 3.695276382000266, 4.407632517289687, 5.230263512348078, 5.421400488859558, 6.201227035762985, 6.826358166142036, 7.350926442161510, 7.853549260518869, 8.440886089941148, 8.616070600930471, 9.353708207724643, 10.00804554251720, 10.26912410283752, 10.71683579347027, 11.58147146738998, 11.79028692896229, 12.40990392217732, 12.94945937579698, 13.32835732248895, 13.97237809901606

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