Properties

Label 2-38640-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s + 9-s − 4·13-s + 15-s − 2·19-s + 21-s − 23-s + 25-s − 27-s + 6·29-s − 2·31-s + 35-s − 10·37-s + 4·39-s + 6·41-s + 4·43-s − 45-s − 6·47-s + 49-s − 6·53-s + 2·57-s − 12·59-s − 10·61-s − 63-s + 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s + 0.258·15-s − 0.458·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.169·35-s − 1.64·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 0.875·47-s + 1/7·49-s − 0.824·53-s + 0.264·57-s − 1.56·59-s − 1.28·61-s − 0.125·63-s + 0.496·65-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: \(0\)
Selberg data: \((2,\ 38640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5032290657\)
\(L(\frac12)\) \(\approx\) \(0.5032290657\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.88987300060495, −14.17418416777912, −13.99487050081815, −13.07731041044436, −12.57227580537729, −12.26025014861563, −11.83418772591279, −11.05551246024644, −10.72485477874790, −10.09500288543324, −9.620699707776594, −9.014226455580490, −8.401199211146469, −7.674654793479019, −7.329118820096355, −6.574147647629952, −6.241204794186726, −5.425489132717972, −4.827063279507299, −4.402336437017189, −3.603998681042547, −2.949396522748645, −2.201536921377789, −1.329300235107905, −0.2767077199927733, 0.2767077199927733, 1.329300235107905, 2.201536921377789, 2.949396522748645, 3.603998681042547, 4.402336437017189, 4.827063279507299, 5.425489132717972, 6.241204794186726, 6.574147647629952, 7.329118820096355, 7.674654793479019, 8.401199211146469, 9.014226455580490, 9.620699707776594, 10.09500288543324, 10.72485477874790, 11.05551246024644, 11.83418772591279, 12.26025014861563, 12.57227580537729, 13.07731041044436, 13.99487050081815, 14.17418416777912, 14.88987300060495

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