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Properties

Label 2-72450-1.1-c1-0-19
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s + 11-s + 2·13-s + 14-s + 16-s − 6·17-s + 5·19-s − 22-s − 23-s − 2·26-s − 28-s − 8·29-s + 4·31-s − 32-s + 6·34-s + 6·37-s − 5·38-s + 11·41-s + 2·43-s + 44-s + 46-s − 3·47-s + 49-s + 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 1.14·19-s − 0.213·22-s − 0.208·23-s − 0.392·26-s − 0.188·28-s − 1.48·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.986·37-s − 0.811·38-s + 1.71·41-s + 0.304·43-s + 0.150·44-s + 0.147·46-s − 0.437·47-s + 1/7·49-s + 0.277·52-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: \(0\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.542726729\)
\(L(\frac12)\) \(\approx\) \(1.542726729\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.07670404024910, −13.45539025701685, −13.22169347177304, −12.58615500909114, −11.98253487480226, −11.45419279937778, −11.04417773288047, −10.71616787521467, −9.826175831948530, −9.530726354533786, −9.160027261826778, −8.494709096596969, −8.067236679495057, −7.412302025762816, −6.886878909249254, −6.510537491340142, −5.650818252996850, −5.546156262635734, −4.292382335180900, −4.121070652893595, −3.240197572938230, −2.605126924963585, −2.016273888813850, −1.161114365359734, −0.5081002832933931, 0.5081002832933931, 1.161114365359734, 2.016273888813850, 2.605126924963585, 3.240197572938230, 4.121070652893595, 4.292382335180900, 5.546156262635734, 5.650818252996850, 6.510537491340142, 6.886878909249254, 7.412302025762816, 8.067236679495057, 8.494709096596969, 9.160027261826778, 9.530726354533786, 9.826175831948530, 10.71616787521467, 11.04417773288047, 11.45419279937778, 11.98253487480226, 12.58615500909114, 13.22169347177304, 13.45539025701685, 14.07670404024910

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