Properties

Label 2-37440-1.1-c1-0-35
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·7-s + 4·11-s + 13-s + 4·17-s − 6·19-s + 25-s − 4·29-s + 6·31-s − 2·35-s + 2·37-s − 10·41-s − 8·43-s − 3·49-s − 4·53-s − 4·55-s + 4·59-s − 2·61-s − 65-s − 6·67-s + 8·71-s + 10·73-s + 8·77-s − 4·79-s − 12·83-s − 4·85-s + 2·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s + 1.20·11-s + 0.277·13-s + 0.970·17-s − 1.37·19-s + 1/5·25-s − 0.742·29-s + 1.07·31-s − 0.338·35-s + 0.328·37-s − 1.56·41-s − 1.21·43-s − 3/7·49-s − 0.549·53-s − 0.539·55-s + 0.520·59-s − 0.256·61-s − 0.124·65-s − 0.733·67-s + 0.949·71-s + 1.17·73-s + 0.911·77-s − 0.450·79-s − 1.31·83-s − 0.433·85-s + 0.211·89-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: \(0\)
Selberg data: \((2,\ 37440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.416320984\)
\(L(\frac12)\) \(\approx\) \(2.416320984\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−14.80714309705387, −14.46037629351189, −13.93707085453610, −13.30426709648742, −12.74164443615972, −12.09410957874151, −11.75739014647127, −11.25021674656278, −10.77521322980951, −10.03480237246845, −9.647516767834423, −8.820341854764429, −8.372580739283517, −8.054210072743076, −7.274839651367673, −6.666693028222297, −6.228952997749889, −5.483562051741089, −4.750446899477233, −4.308428951852800, −3.609054969754877, −3.103807097014469, −1.964917190790346, −1.515935578786788, −0.5913301424842526, 0.5913301424842526, 1.515935578786788, 1.964917190790346, 3.103807097014469, 3.609054969754877, 4.308428951852800, 4.750446899477233, 5.483562051741089, 6.228952997749889, 6.666693028222297, 7.274839651367673, 8.054210072743076, 8.372580739283517, 8.820341854764429, 9.647516767834423, 10.03480237246845, 10.77521322980951, 11.25021674656278, 11.75739014647127, 12.09410957874151, 12.74164443615972, 13.30426709648742, 13.93707085453610, 14.46037629351189, 14.80714309705387

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