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Properties

Label 2-72450-1.1-c1-0-70
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 − 6·11-s + 2·13-s − 14-s + 16-s + 6·19-s + 6·22-s − 23-s − 2·26-s + 28-s + 4·29-s + 2·31-s − 32-s + 6·37-s − 6·38-s − 6·41-s − 6·43-s − 6·44-s + 46-s + 2·47-s + 49-s + 2·52-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 − 1.80·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.37·19-s + 1.27·22-s − 0.208·23-s − 0.392·26-s + 0.188·28-s + 0.742·29-s + 0.359·31-s − 0.176·32-s + 0.986·37-s − 0.973·38-s − 0.937·41-s − 0.914·43-s − 0.904·44-s + 0.147·46-s + 0.291·47-s + 1/7·49-s + 0.277·52-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.26605031705677, −13.85267984181038, −13.33974084263688, −12.90310198384671, −12.28171003433200, −11.64383857302518, −11.39110475145502, −10.69771837528891, −10.27205605809224, −9.943441701394601, −9.296062919672874, −8.674974272112278, −8.134925815048788, −7.841178092411332, −7.340333962703161, −6.701755065593252, −6.023097457621845, −5.485845870556795, −4.981898138001175, −4.409765752950854, −3.414878697959802, −2.957314478101475, −2.369437055300432, −1.567668168741001, −0.8744633182005898, 0, 0.8744633182005898, 1.567668168741001, 2.369437055300432, 2.957314478101475, 3.414878697959802, 4.409765752950854, 4.981898138001175, 5.485845870556795, 6.023097457621845, 6.701755065593252, 7.340333962703161, 7.841178092411332, 8.134925815048788, 8.674974272112278, 9.296062919672874, 9.943441701394601, 10.27205605809224, 10.69771837528891, 11.39110475145502, 11.64383857302518, 12.28171003433200, 12.90310198384671, 13.33974084263688, 13.85267984181038, 14.26605031705677

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