Properties

Label 2-72450-1.1-c1-0-58
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 + 14-s + 16-s + 6·17-s + 4·22-s − 23-s − 28-s + 8·29-s − 8·31-s − 32-s − 6·34-s + 2·37-s − 2·41-s − 8·43-s − 4·44-s + 46-s + 49-s + 56-s − 8·58-s − 10·59-s + 8·62-s + 64-s + 12·67-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.20·11-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.852·22-s − 0.208·23-s − 0.188·28-s + 1.48·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.312·41-s − 1.21·43-s − 0.603·44-s + 0.147·46-s + 1/7·49-s + 0.133·56-s − 1.05·58-s − 1.30·59-s + 1.01·62-s + 1/8·64-s + 1.46·67-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: $\chi_{72450} (1, \cdot )$
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 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
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 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.38813445015829, −13.86610565060580, −13.32320716955819, −12.74052426756904, −12.31542471539410, −11.92803449199394, −11.19033043194696, −10.71725285951048, −10.27479551329028, −9.787886600524276, −9.466293441225762, −8.662726987204276, −8.174654277098522, −7.839380387848225, −7.222252126302715, −6.732097993788414, −6.049703665206864, −5.497031758940894, −5.065805937331373, −4.283401022698161, −3.347077294793955, −3.104146456029607, −2.326300752424492, −1.615175892375658, −0.7956285151624908, 0, 0.7956285151624908, 1.615175892375658, 2.326300752424492, 3.104146456029607, 3.347077294793955, 4.283401022698161, 5.065805937331373, 5.497031758940894, 6.049703665206864, 6.732097993788414, 7.222252126302715, 7.839380387848225, 8.174654277098522, 8.662726987204276, 9.466293441225762, 9.787886600524276, 10.27479551329028, 10.71725285951048, 11.19033043194696, 11.92803449199394, 12.31542471539410, 12.74052426756904, 13.32320716955819, 13.86610565060580, 14.38813445015829

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