Properties

Label 2-92400-1.1-c1-0-122
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 11-s + 2·13-s + 2·17-s + 21-s − 27-s − 2·29-s + 4·31-s + 33-s − 6·37-s − 2·39-s + 6·41-s + 12·43-s + 4·47-s + 49-s − 2·51-s − 6·53-s + 4·59-s − 10·61-s − 63-s − 4·67-s − 8·71-s + 2·73-s + 77-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.485·17-s + 0.218·21-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.174·33-s − 0.986·37-s − 0.320·39-s + 0.937·41-s + 1.82·43-s + 0.583·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.520·59-s − 1.28·61-s − 0.125·63-s − 0.488·67-s − 0.949·71-s + 0.234·73-s + 0.113·77-s + 1/9·81-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.12967287907512, −13.56039568475770, −12.97855067611371, −12.63465523694006, −12.13291644158036, −11.67220369917871, −11.02458311177528, −10.71787417365258, −10.18398579223322, −9.692725497259872, −9.108372354085203, −8.662658684044453, −7.995063905985202, −7.394180571083402, −7.107906313130052, −6.234051791391072, −5.952525316523978, −5.503152844490989, −4.730938516591213, −4.273422535963986, −3.609642311280191, −2.998258997288927, −2.355877027887232, −1.486510614646486, −0.8545146071009099, 0, 0.8545146071009099, 1.486510614646486, 2.355877027887232, 2.998258997288927, 3.609642311280191, 4.273422535963986, 4.730938516591213, 5.503152844490989, 5.952525316523978, 6.234051791391072, 7.107906313130052, 7.394180571083402, 7.995063905985202, 8.662658684044453, 9.108372354085203, 9.692725497259872, 10.18398579223322, 10.71787417365258, 11.02458311177528, 11.67220369917871, 12.13291644158036, 12.63465523694006, 12.97855067611371, 13.56039568475770, 14.12967287907512

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