↑

Properties

Label 2-92400-1.1-c1-0-111
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 + 13-s − 5·19-s − 21-s + 2·23-s − 27-s − 29-s − 8·31-s + 33-s − 37-s − 39-s + 6·43-s + 47-s + 49-s + 2·53-s + 5·57-s − 9·59-s + 10·61-s + 63-s + 7·67-s − 2·69-s − 9·73-s − 77-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 1.14·19-s − 0.218·21-s + 0.417·23-s − 0.192·27-s − 0.185·29-s − 1.43·31-s + 0.174·33-s − 0.164·37-s − 0.160·39-s + 0.914·43-s + 0.145·47-s + 1/7·49-s + 0.274·53-s + 0.662·57-s − 1.17·59-s + 1.28·61-s + 0.125·63-s + 0.855·67-s − 0.240·69-s − 1.05·73-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: Trivial
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 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−14.25661119293644, −13.37241109889679, −13.14867218792845, −12.49209791913539, −12.29018544435988, −11.45767006608618, −11.09752652100679, −10.78186013910029, −10.23430431511547, −9.687691570345567, −8.997676671202397, −8.674439351383224, −8.032945838916235, −7.447575138286666, −7.025673756316878, −6.402714863037300, −5.841836433272543, −5.399341145465750, −4.821960257483763, −4.208253800836935, −3.749920439849383, −2.953917288549394, −2.188702665956755, −1.662582041144477, −0.8088308134644657, 0, 0.8088308134644657, 1.662582041144477, 2.188702665956755, 2.953917288549394, 3.749920439849383, 4.208253800836935, 4.821960257483763, 5.399341145465750, 5.841836433272543, 6.402714863037300, 7.025673756316878, 7.447575138286666, 8.032945838916235, 8.674439351383224, 8.997676671202397, 9.687691570345567, 10.23430431511547, 10.78186013910029, 11.09752652100679, 11.45767006608618, 12.29018544435988, 12.49209791913539, 13.14867218792845, 13.37241109889679, 14.25661119293644

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