↑

Properties

Label 2-72450-1.1-c1-0-116
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 − 14-s + 16-s + 6·19-s − 4·22-s + 23-s + 28-s + 10·29-s − 4·31-s − 32-s − 10·37-s − 6·38-s + 2·41-s − 4·43-s + 4·44-s − 46-s + 49-s − 4·53-s − 56-s − 10·58-s − 6·59-s + 6·61-s + 4·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1.20·11-s − 0.267·14-s + 1/4·16-s + 1.37·19-s − 0.852·22-s + 0.208·23-s + 0.188·28-s + 1.85·29-s − 0.718·31-s − 0.176·32-s − 1.64·37-s − 0.973·38-s + 0.312·41-s − 0.609·43-s + 0.603·44-s − 0.147·46-s + 1/7·49-s − 0.549·53-s − 0.133·56-s − 1.31·58-s − 0.781·59-s + 0.768·61-s + 0.508·62-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 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
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

−14.35510645496393, −13.85092779981149, −13.62336064129435, −12.50585942836510, −12.37247724011938, −11.70852511135089, −11.40298613148895, −10.83747906882879, −10.21654006074447, −9.794906153078394, −9.268214763021268, −8.717064231519427, −8.410710467914437, −7.668542855777691, −7.194794216026391, −6.699301287371969, −6.210426085690001, −5.456894776706786, −4.962542158566294, −4.274447391421558, −3.520735215110637, −3.069100465183584, −2.249353581967254, −1.364115703612057, −1.140127893083031, 0, 1.140127893083031, 1.364115703612057, 2.249353581967254, 3.069100465183584, 3.520735215110637, 4.274447391421558, 4.962542158566294, 5.456894776706786, 6.210426085690001, 6.699301287371969, 7.194794216026391, 7.668542855777691, 8.410710467914437, 8.717064231519427, 9.268214763021268, 9.794906153078394, 10.21654006074447, 10.83747906882879, 11.40298613148895, 11.70852511135089, 12.37247724011938, 12.50585942836510, 13.62336064129435, 13.85092779981149, 14.35510645496393

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