Properties

Label 2-38640-1.1-c1-0-11
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 + 2·11-s − 6·13-s − 15-s − 4·17-s + 6·19-s − 21-s − 23-s + 25-s − 27-s + 6·31-s − 2·33-s + 35-s + 6·37-s + 6·39-s − 6·41-s − 6·43-s + 45-s − 2·47-s + 49-s + 4·51-s + 2·53-s + 2·55-s − 6·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s − 0.258·15-s − 0.970·17-s + 1.37·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.07·31-s − 0.348·33-s + 0.169·35-s + 0.986·37-s + 0.960·39-s − 0.937·41-s − 0.914·43-s + 0.149·45-s − 0.291·47-s + 1/7·49-s + 0.560·51-s + 0.274·53-s + 0.269·55-s − 0.794·57-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: \(0\)
Selberg data: \((2,\ 38640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.780610117\)
\(L(\frac12)\) \(\approx\) \(1.780610117\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.94117329253442, −14.25499822388410, −13.81322062985891, −13.27981792348021, −12.73603990813344, −11.99527923107498, −11.70098080147772, −11.42743340366904, −10.41482126570725, −10.18422155342531, −9.540695734918926, −9.156251711730422, −8.418223020470665, −7.689933547605365, −7.249984679391328, −6.632441112804009, −6.147789742289110, −5.411798604325454, −4.789075300795406, −4.592806318439900, −3.613444645701296, −2.799192648052935, −2.145955412568663, −1.402353289897110, −0.5198899682293443, 0.5198899682293443, 1.402353289897110, 2.145955412568663, 2.799192648052935, 3.613444645701296, 4.592806318439900, 4.789075300795406, 5.411798604325454, 6.147789742289110, 6.632441112804009, 7.249984679391328, 7.689933547605365, 8.418223020470665, 9.156251711730422, 9.540695734918926, 10.18422155342531, 10.41482126570725, 11.42743340366904, 11.70098080147772, 11.99527923107498, 12.73603990813344, 13.27981792348021, 13.81322062985891, 14.25499822388410, 14.94117329253442

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