Properties

Label 2-38640-1.1-c1-0-36
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 − 2·13-s + 15-s + 6·17-s + 4·19-s + 21-s − 23-s + 25-s − 27-s − 10·29-s + 4·31-s + 35-s + 10·37-s + 2·39-s − 2·41-s + 4·43-s − 45-s + 49-s − 6·51-s − 6·53-s − 4·57-s + 4·59-s − 6·61-s − 63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.169·35-s + 1.64·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.125·63-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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.19541625458464, −14.45853324376123, −14.24259070443557, −13.41453064382431, −12.85555657315232, −12.50487282563986, −11.86315643374227, −11.51087123422274, −11.03361930959881, −10.25197210115568, −9.787049692815118, −9.490719839216130, −8.706816699984677, −7.899262506317822, −7.509020534829574, −7.158218747763234, −6.255945333887347, −5.757769553342376, −5.317365917457048, −4.541481484139377, −3.980177381698039, −3.237048433770309, −2.724170525731764, −1.638285750493973, −0.8945132471802122, 0, 0.8945132471802122, 1.638285750493973, 2.724170525731764, 3.237048433770309, 3.980177381698039, 4.541481484139377, 5.317365917457048, 5.757769553342376, 6.255945333887347, 7.158218747763234, 7.509020534829574, 7.899262506317822, 8.706816699984677, 9.490719839216130, 9.787049692815118, 10.25197210115568, 11.03361930959881, 11.51087123422274, 11.86315643374227, 12.50487282563986, 12.85555657315232, 13.41453064382431, 14.24259070443557, 14.45853324376123, 15.19541625458464

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