Properties

Label 2-37440-1.1-c1-0-143
Degree $2$
Conductor $37440$
Sign $-1$
Analytic cond. $298.959$
Root an. cond. $17.2904$
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  + 5-s + 4·7-s + 4·11-s − 13-s − 6·17-s + 4·23-s + 25-s − 6·29-s − 8·31-s + 4·35-s + 2·37-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s − 2·53-s + 4·55-s + 4·59-s − 14·61-s − 65-s + 12·67-s + 8·71-s − 10·73-s + 16·77-s − 4·83-s − 6·85-s − 10·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 1.20·11-s − 0.277·13-s − 1.45·17-s + 0.834·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.274·53-s + 0.539·55-s + 0.520·59-s − 1.79·61-s − 0.124·65-s + 1.46·67-s + 0.949·71-s − 1.17·73-s + 1.82·77-s − 0.439·83-s − 0.650·85-s − 1.05·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37440\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(298.959\)
Root analytic conductor: \(17.2904\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{37440} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 37440,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
13 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−14.91686921399812, −14.74332168967363, −14.07911614599567, −13.72447331892035, −12.96451414804188, −12.67430527947973, −11.66284462151861, −11.55684718454248, −10.92703796354613, −10.63146809094561, −9.614391973643196, −9.266066703654967, −8.764369281650985, −8.238540708334308, −7.561938977297054, −6.907538056080750, −6.578376872761616, −5.675171490661097, −5.188621148471912, −4.585617617970808, −4.075796291934814, −3.321956528613692, −2.326594826847864, −1.741452146632938, −1.298866783328736, 0, 1.298866783328736, 1.741452146632938, 2.326594826847864, 3.321956528613692, 4.075796291934814, 4.585617617970808, 5.188621148471912, 5.675171490661097, 6.578376872761616, 6.907538056080750, 7.561938977297054, 8.238540708334308, 8.764369281650985, 9.266066703654967, 9.614391973643196, 10.63146809094561, 10.92703796354613, 11.55684718454248, 11.66284462151861, 12.67430527947973, 12.96451414804188, 13.72447331892035, 14.07911614599567, 14.74332168967363, 14.91686921399812

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