Properties

Label 2-92400-1.1-c1-0-12
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s − 11-s − 3·13-s + 17-s − 7·19-s + 21-s − 4·23-s + 27-s − 8·29-s + 2·31-s − 33-s + 11·37-s − 3·39-s − 9·41-s − 8·43-s − 4·47-s + 49-s + 51-s + 5·53-s − 7·57-s − 4·59-s + 61-s + 63-s − 11·67-s − 4·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.242·17-s − 1.60·19-s + 0.218·21-s − 0.834·23-s + 0.192·27-s − 1.48·29-s + 0.359·31-s − 0.174·33-s + 1.80·37-s − 0.480·39-s − 1.40·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.140·51-s + 0.686·53-s − 0.927·57-s − 0.520·59-s + 0.128·61-s + 0.125·63-s − 1.34·67-s − 0.481·69-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

\(L(1)\) \(\approx\) \(1.427235204\)
\(L(\frac12)\) \(\approx\) \(1.427235204\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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

−13.74921400040360, −13.38029312089609, −12.95672161141383, −12.35466054321053, −12.00360305877930, −11.29437686378940, −10.92881874865964, −10.26162897270213, −9.821021414898822, −9.485764894025607, −8.656381331134307, −8.390647668276367, −7.753727673293599, −7.508343296789058, −6.648176490460427, −6.340527692486789, −5.521791257540527, −5.044163503677359, −4.367813964882673, −3.978897380048599, −3.252549925799615, −2.543391500820154, −2.046824564473661, −1.507928283670131, −0.3429964692559481, 0.3429964692559481, 1.507928283670131, 2.046824564473661, 2.543391500820154, 3.252549925799615, 3.978897380048599, 4.367813964882673, 5.044163503677359, 5.521791257540527, 6.340527692486789, 6.648176490460427, 7.508343296789058, 7.753727673293599, 8.390647668276367, 8.656381331134307, 9.485764894025607, 9.821021414898822, 10.26162897270213, 10.92881874865964, 11.29437686378940, 12.00360305877930, 12.35466054321053, 12.95672161141383, 13.38029312089609, 13.74921400040360

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