↑

Properties

Label 2-92400-1.1-c1-0-116
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 + 5·13-s + 3·17-s − 8·19-s + 21-s + 7·23-s − 27-s − 5·29-s − 7·31-s + 33-s − 4·37-s − 5·39-s + 3·41-s + 9·43-s + 2·47-s + 49-s − 3·51-s − 3·53-s + 8·57-s − 5·59-s − 9·61-s − 63-s + 12·67-s − 7·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.38·13-s + 0.727·17-s − 1.83·19-s + 0.218·21-s + 1.45·23-s − 0.192·27-s − 0.928·29-s − 1.25·31-s + 0.174·33-s − 0.657·37-s − 0.800·39-s + 0.468·41-s + 1.37·43-s + 0.291·47-s + 1/7·49-s − 0.420·51-s − 0.412·53-s + 1.05·57-s − 0.650·59-s − 1.15·61-s − 0.125·63-s + 1.46·67-s − 0.842·69-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: 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 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.98353014113747, −13.50179722412627, −12.94967056341190, −12.65034617789034, −12.33079527724806, −11.46854736238951, −10.94537899032590, −10.79326243197507, −10.37827498912686, −9.539778496185924, −8.989847002966610, −8.830126421035286, −7.917567669817518, −7.594339667956376, −6.823889101409108, −6.454517395236696, −5.866100907866685, −5.480821064846044, −4.839476375212147, −4.098591518321580, −3.690784118676355, −3.070432188464826, −2.243679924274245, −1.558597118351647, −0.8306561488399215, 0, 0.8306561488399215, 1.558597118351647, 2.243679924274245, 3.070432188464826, 3.690784118676355, 4.098591518321580, 4.839476375212147, 5.480821064846044, 5.866100907866685, 6.454517395236696, 6.823889101409108, 7.594339667956376, 7.917567669817518, 8.830126421035286, 8.989847002966610, 9.539778496185924, 10.37827498912686, 10.79326243197507, 10.94537899032590, 11.46854736238951, 12.33079527724806, 12.65034617789034, 12.94967056341190, 13.50179722412627, 13.98353014113747

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