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Properties

Label 2-72450-1.1-c1-0-50
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 − 4·11-s − 6·13-s + 14-s + 16-s + 6·19-s + 4·22-s + 23-s + 6·26-s − 28-s + 8·29-s + 8·31-s − 32-s − 2·37-s − 6·38-s − 2·41-s − 8·43-s − 4·44-s − 46-s − 12·47-s + 49-s − 6·52-s + 2·53-s + 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.37·19-s + 0.852·22-s + 0.208·23-s + 1.17·26-s − 0.188·28-s + 1.48·29-s + 1.43·31-s − 0.176·32-s − 0.328·37-s − 0.973·38-s − 0.312·41-s − 1.21·43-s − 0.603·44-s − 0.147·46-s − 1.75·47-s + 1/7·49-s − 0.832·52-s + 0.274·53-s + 0.133·56-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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.40441058673215, −13.85572530545899, −13.25471002100246, −12.91885893173822, −12.09449860251342, −11.89768814894117, −11.47752395237934, −10.51833914659077, −10.24854967851989, −9.826150185967117, −9.517297197710639, −8.675387677738176, −8.171218396939424, −7.819170008152892, −7.075654331475360, −6.890834229087241, −6.130035747783478, −5.363561487588025, −4.974000770646960, −4.490258553065326, −3.338664898471079, −2.888541318869359, −2.503523073149804, −1.618943295695724, −0.7340751005446500, 0, 0.7340751005446500, 1.618943295695724, 2.503523073149804, 2.888541318869359, 3.338664898471079, 4.490258553065326, 4.974000770646960, 5.363561487588025, 6.130035747783478, 6.890834229087241, 7.075654331475360, 7.819170008152892, 8.171218396939424, 8.675387677738176, 9.517297197710639, 9.826150185967117, 10.24854967851989, 10.51833914659077, 11.47752395237934, 11.89768814894117, 12.09449860251342, 12.91885893173822, 13.25471002100246, 13.85572530545899, 14.40441058673215

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