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Properties

Label 2-72450-1.1-c1-0-108
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s + 2·11-s + 6·13-s + 14-s + 16-s + 6·17-s − 2·22-s − 23-s − 6·26-s − 28-s − 4·29-s − 2·31-s − 32-s − 6·34-s − 4·37-s − 2·41-s + 4·43-s + 2·44-s + 46-s + 49-s + 6·52-s − 6·53-s + 56-s + 4·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.603·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.426·22-s − 0.208·23-s − 1.17·26-s − 0.188·28-s − 0.742·29-s − 0.359·31-s − 0.176·32-s − 1.02·34-s − 0.657·37-s − 0.312·41-s + 0.609·43-s + 0.301·44-s + 0.147·46-s + 1/7·49-s + 0.832·52-s − 0.824·53-s + 0.133·56-s + 0.525·58-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.41518360382595, −13.84571303881285, −13.39861634570223, −12.76328501515016, −12.29069676671357, −11.80019864974763, −11.26921965988576, −10.75390036096385, −10.37197206230926, −9.679836794259875, −9.336005881921400, −8.735330535633651, −8.377072846830241, −7.610106253478582, −7.380966129052682, −6.448530058703213, −6.212612923603248, −5.651656041027451, −5.011205327050607, −4.044527024095471, −3.528568958664209, −3.212262049614905, −2.219550465073416, −1.426084683430717, −1.047028811458369, 0, 1.047028811458369, 1.426084683430717, 2.219550465073416, 3.212262049614905, 3.528568958664209, 4.044527024095471, 5.011205327050607, 5.651656041027451, 6.212612923603248, 6.448530058703213, 7.380966129052682, 7.610106253478582, 8.377072846830241, 8.735330535633651, 9.336005881921400, 9.679836794259875, 10.37197206230926, 10.75390036096385, 11.26921965988576, 11.80019864974763, 12.29069676671357, 12.76328501515016, 13.39861634570223, 13.84571303881285, 14.41518360382595

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