Properties

Label 2-38640-1.1-c1-0-35
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 − 4·11-s + 6·13-s + 15-s − 4·17-s − 21-s − 23-s + 25-s − 27-s + 4·33-s − 35-s − 6·37-s − 6·39-s − 45-s − 8·47-s + 49-s + 4·51-s + 8·53-s + 4·55-s + 10·61-s + 63-s − 6·65-s + 8·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s − 0.970·17-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.696·33-s − 0.169·35-s − 0.986·37-s − 0.960·39-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.560·51-s + 1.09·53-s + 0.539·55-s + 1.28·61-s + 0.125·63-s − 0.744·65-s + 0.977·67-s + 0.120·69-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−15.25707617633591, −14.67016601800357, −13.84578819254819, −13.54465644845057, −12.89591966873907, −12.65468233053367, −11.71049346234241, −11.43597597998910, −10.93419223679157, −10.49054990717020, −10.01523116555286, −9.148091674686286, −8.510236127972320, −8.246265823779667, −7.589530016879951, −6.861890593956237, −6.463911255499914, −5.681062512756339, −5.253983776330580, −4.607867341146296, −3.934827699482452, −3.411049469442680, −2.486988355023541, −1.762831742078469, −0.8795912219916253, 0, 0.8795912219916253, 1.762831742078469, 2.486988355023541, 3.411049469442680, 3.934827699482452, 4.607867341146296, 5.253983776330580, 5.681062512756339, 6.463911255499914, 6.861890593956237, 7.589530016879951, 8.246265823779667, 8.510236127972320, 9.148091674686286, 10.01523116555286, 10.49054990717020, 10.93419223679157, 11.43597597998910, 11.71049346234241, 12.65468233053367, 12.89591966873907, 13.54465644845057, 13.84578819254819, 14.67016601800357, 15.25707617633591

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