Properties

Label 2-38640-1.1-c1-0-45
Degree $2$
Conductor $38640$
Sign $-1$
Analytic cond. $308.541$
Root an. cond. $17.5653$
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  − 3-s − 5-s + 7-s + 9-s − 11-s + 15-s + 2·17-s + 3·19-s − 21-s − 23-s + 25-s − 27-s − 6·29-s + 33-s − 35-s + 6·37-s − 3·41-s + 6·43-s − 45-s + 7·47-s + 49-s − 2·51-s − 7·53-s + 55-s − 3·57-s + 3·59-s + 61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.258·15-s + 0.485·17-s + 0.688·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.174·33-s − 0.169·35-s + 0.986·37-s − 0.468·41-s + 0.914·43-s − 0.149·45-s + 1.02·47-s + 1/7·49-s − 0.280·51-s − 0.961·53-s + 0.134·55-s − 0.397·57-s + 0.390·59-s + 0.128·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38640\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(308.541\)
Root analytic conductor: \(17.5653\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{38640} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 38640,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 + T \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.03236287779176, −14.69529864271143, −14.03119877399595, −13.53232758691366, −12.87300332495058, −12.47448674110126, −11.85082616246453, −11.45192812721674, −10.99165042179700, −10.42169041418028, −9.869158962058448, −9.280411274208827, −8.713997029863606, −7.891478458504151, −7.631206252668718, −7.077029325809587, −6.342491289914850, −5.588554040088367, −5.398044866418148, −4.478025308155031, −4.092562282908983, −3.279216241827970, −2.597565957252332, −1.666316735787010, −0.9377271021404804, 0, 0.9377271021404804, 1.666316735787010, 2.597565957252332, 3.279216241827970, 4.092562282908983, 4.478025308155031, 5.398044866418148, 5.588554040088367, 6.342491289914850, 7.077029325809587, 7.631206252668718, 7.891478458504151, 8.713997029863606, 9.280411274208827, 9.869158962058448, 10.42169041418028, 10.99165042179700, 11.45192812721674, 11.85082616246453, 12.47448674110126, 12.87300332495058, 13.53232758691366, 14.03119877399595, 14.69529864271143, 15.03236287779176

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