Properties

Label 2-37440-1.1-c1-0-147
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s − 4·11-s + 13-s − 8·17-s − 6·19-s + 6·23-s + 25-s − 4·29-s + 2·35-s + 2·37-s + 2·41-s − 4·43-s − 3·49-s − 10·53-s + 4·55-s − 4·59-s + 10·61-s − 65-s + 12·67-s − 8·71-s − 8·73-s + 8·77-s − 8·79-s − 12·83-s + 8·85-s + 14·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 1.20·11-s + 0.277·13-s − 1.94·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s − 0.742·29-s + 0.338·35-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 3/7·49-s − 1.37·53-s + 0.539·55-s − 0.520·59-s + 1.28·61-s − 0.124·65-s + 1.46·67-s − 0.949·71-s − 0.936·73-s + 0.911·77-s − 0.900·79-s − 1.31·83-s + 0.867·85-s + 1.48·89-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: \(2\)
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−15.48404912165736, −14.91594273986733, −14.54565924825730, −13.53303917240664, −13.19678611861856, −12.87555022622779, −12.53098177605622, −11.51450138480258, −11.13245660079234, −10.75722237541275, −10.21095152828580, −9.449631864224118, −8.999541288978012, −8.410598880338988, −7.982754648705660, −7.166058154711334, −6.702612497505525, −6.280532979048818, −5.465664475869840, −4.823401391522860, −4.266434862978878, −3.645924844909229, −2.766420520666157, −2.437682076504312, −1.438782545727172, 0, 0, 1.438782545727172, 2.437682076504312, 2.766420520666157, 3.645924844909229, 4.266434862978878, 4.823401391522860, 5.465664475869840, 6.280532979048818, 6.702612497505525, 7.166058154711334, 7.982754648705660, 8.410598880338988, 8.999541288978012, 9.449631864224118, 10.21095152828580, 10.75722237541275, 11.13245660079234, 11.51450138480258, 12.53098177605622, 12.87555022622779, 13.19678611861856, 13.53303917240664, 14.54565924825730, 14.91594273986733, 15.48404912165736

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