↑

Properties

Label 2-72450-1.1-c1-0-105
Degree $2$
Conductor $72450$
Sign $-1$
Analytic cond. $578.516$
Root an. cond. $24.0523$
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  − 2-s + 4-s − 7-s − 8-s + 4·11-s + 4·13-s + 14-s + 16-s − 4·17-s − 2·19-s − 4·22-s + 23-s − 4·26-s − 28-s + 6·29-s + 6·31-s − 32-s + 4·34-s + 2·37-s + 2·38-s + 10·41-s − 8·43-s + 4·44-s − 46-s + 10·47-s + 49-s + 4·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.20·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.458·19-s − 0.852·22-s + 0.208·23-s − 0.784·26-s − 0.188·28-s + 1.11·29-s + 1.07·31-s − 0.176·32-s + 0.685·34-s + 0.328·37-s + 0.324·38-s + 1.56·41-s − 1.21·43-s + 0.603·44-s − 0.147·46-s + 1.45·47-s + 1/7·49-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(578.516\)
Root analytic conductor: \(24.0523\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−14.45230912827286, −13.69315509936248, −13.51103439797913, −12.78869548634901, −12.25907935252844, −11.76451511930842, −11.30045491899865, −10.75086021156749, −10.41794244695247, −9.636820240760629, −9.308197192326089, −8.681787876744943, −8.468914263636009, −7.801721288227873, −7.022613893110160, −6.591049826023479, −6.250251254047605, −5.741404549628625, −4.764732194100748, −4.163444154525596, −3.754408267501814, −2.849965749539062, −2.418684365361935, −1.378145684702103, −1.044112777468182, 0, 1.044112777468182, 1.378145684702103, 2.418684365361935, 2.849965749539062, 3.754408267501814, 4.163444154525596, 4.764732194100748, 5.741404549628625, 6.250251254047605, 6.591049826023479, 7.022613893110160, 7.801721288227873, 8.468914263636009, 8.681787876744943, 9.308197192326089, 9.636820240760629, 10.41794244695247, 10.75086021156749, 11.30045491899865, 11.76451511930842, 12.25907935252844, 12.78869548634901, 13.51103439797913, 13.69315509936248, 14.45230912827286

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