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Properties

Label 2-72450-1.1-c1-0-6
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 − 2·13-s + 14-s + 16-s − 4·19-s − 23-s + 2·26-s − 28-s − 4·31-s − 32-s − 2·37-s + 4·38-s − 12·41-s + 4·43-s + 46-s + 12·47-s + 49-s − 2·52-s + 56-s − 12·59-s + 14·61-s + 4·62-s + 64-s + 4·67-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.208·23-s + 0.392·26-s − 0.188·28-s − 0.718·31-s − 0.176·32-s − 0.328·37-s + 0.648·38-s − 1.87·41-s + 0.609·43-s + 0.147·46-s + 1.75·47-s + 1/7·49-s − 0.277·52-s + 0.133·56-s − 1.56·59-s + 1.79·61-s + 0.508·62-s + 1/8·64-s + 0.488·67-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

\(L(1)\) \(\approx\) \(0.7781054153\)
\(L(\frac12)\) \(\approx\) \(0.7781054153\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.96651943852437, −13.79007244498050, −12.90174004277831, −12.56957459994973, −12.13633468574282, −11.54447715418256, −10.94489208271236, −10.56663453632012, −9.988966371822588, −9.616075070641749, −8.966095502258266, −8.590178643610379, −8.030275211414995, −7.379299717795882, −6.993063432941739, −6.406376712817807, −5.891643963280110, −5.207969552652595, −4.656932431756428, −3.772556039487084, −3.435157234724609, −2.379206882871504, −2.186271322101696, −1.214829020289346, −0.3401645882497650, 0.3401645882497650, 1.214829020289346, 2.186271322101696, 2.379206882871504, 3.435157234724609, 3.772556039487084, 4.656932431756428, 5.207969552652595, 5.891643963280110, 6.406376712817807, 6.993063432941739, 7.379299717795882, 8.030275211414995, 8.590178643610379, 8.966095502258266, 9.616075070641749, 9.988966371822588, 10.56663453632012, 10.94489208271236, 11.54447715418256, 12.13633468574282, 12.56957459994973, 12.90174004277831, 13.79007244498050, 13.96651943852437

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