Properties

Label 2-38640-1.1-c1-0-43
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 + 2·13-s − 15-s + 4·17-s + 4·19-s + 21-s − 23-s + 25-s − 27-s − 8·29-s − 8·31-s + 4·33-s − 35-s − 2·37-s − 2·39-s + 4·41-s + 8·43-s + 45-s + 49-s − 4·51-s − 4·53-s − 4·55-s − 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.970·17-s + 0.917·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s − 1.43·31-s + 0.696·33-s − 0.169·35-s − 0.328·37-s − 0.320·39-s + 0.624·41-s + 1.21·43-s + 0.149·45-s + 1/7·49-s − 0.560·51-s − 0.549·53-s − 0.539·55-s − 0.529·57-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.04750185935834, −14.63636860271357, −13.90118871204778, −13.51296568951764, −12.89313882363070, −12.61697922500485, −12.01488277195534, −11.28872939960968, −10.90065627520101, −10.37495566065627, −9.877439388451892, −9.289532204189474, −8.888209209117142, −7.856755666270529, −7.593912836851322, −7.064851386791750, −6.165513149750528, −5.699454021293128, −5.413493582774373, −4.745703049221520, −3.781685227943303, −3.357402524749737, −2.520517967043766, −1.768797149678281, −0.9318977469920050, 0, 0.9318977469920050, 1.768797149678281, 2.520517967043766, 3.357402524749737, 3.781685227943303, 4.745703049221520, 5.413493582774373, 5.699454021293128, 6.165513149750528, 7.064851386791750, 7.593912836851322, 7.856755666270529, 8.888209209117142, 9.289532204189474, 9.877439388451892, 10.37495566065627, 10.90065627520101, 11.28872939960968, 12.01488277195534, 12.61697922500485, 12.89313882363070, 13.51296568951764, 13.90118871204778, 14.63636860271357, 15.04750185935834

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