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Properties

Label 2-72450-1.1-c1-0-104
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 + 2·13-s + 14-s + 16-s − 2·17-s − 4·19-s − 2·22-s + 23-s + 2·26-s + 28-s + 2·29-s − 8·31-s + 32-s − 2·34-s + 4·37-s − 4·38-s − 12·41-s + 2·43-s − 2·44-s + 46-s + 8·47-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 − 0.603·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.426·22-s + 0.208·23-s + 0.392·26-s + 0.188·28-s + 0.371·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s + 0.657·37-s − 0.648·38-s − 1.87·41-s + 0.304·43-s − 0.301·44-s + 0.147·46-s + 1.16·47-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−14.31014395226433, −13.81796062406667, −13.26888883437426, −13.01047665287183, −12.44175962666153, −11.88749251840951, −11.37866194426247, −10.82455695437900, −10.57724464314808, −9.960169823919883, −9.218017785521855, −8.639742123907711, −8.262135715945949, −7.641390127653948, −6.940891010072074, −6.681381324227719, −5.854399669788801, −5.441644076263607, −4.943855795080762, −4.152916348492545, −3.897011596634435, −3.065981016917263, −2.390144719352299, −1.896480836940851, −1.037231255736845, 0, 1.037231255736845, 1.896480836940851, 2.390144719352299, 3.065981016917263, 3.897011596634435, 4.152916348492545, 4.943855795080762, 5.441644076263607, 5.854399669788801, 6.681381324227719, 6.940891010072074, 7.641390127653948, 8.262135715945949, 8.639742123907711, 9.218017785521855, 9.960169823919883, 10.57724464314808, 10.82455695437900, 11.37866194426247, 11.88749251840951, 12.44175962666153, 13.01047665287183, 13.26888883437426, 13.81796062406667, 14.31014395226433

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