Properties

Label 2-15680-1.1-c1-0-36
Degree $2$
Conductor $15680$
Sign $1$
Analytic cond. $125.205$
Root an. cond. $11.1895$
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  − 3·3-s + 5-s + 6·9-s + 3·11-s + 13-s − 3·15-s + 17-s + 4·19-s + 4·23-s + 25-s − 9·27-s + 9·29-s − 6·31-s − 9·33-s + 8·37-s − 3·39-s − 6·41-s + 8·43-s + 6·45-s + 7·47-s − 3·51-s + 8·53-s + 3·55-s − 12·57-s + 4·59-s + 10·61-s + 65-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 2·9-s + 0.904·11-s + 0.277·13-s − 0.774·15-s + 0.242·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s − 1.07·31-s − 1.56·33-s + 1.31·37-s − 0.480·39-s − 0.937·41-s + 1.21·43-s + 0.894·45-s + 1.02·47-s − 0.420·51-s + 1.09·53-s + 0.404·55-s − 1.58·57-s + 0.520·59-s + 1.28·61-s + 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15680\)    =    \(2^{6} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(125.205\)
Root analytic conductor: \(11.1895\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{15680} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.800626634\)
\(L(\frac12)\) \(\approx\) \(1.800626634\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−16.24456183848685, −15.65953335370342, −14.95357848854886, −14.26758835949399, −13.77505012694321, −12.94638284275690, −12.64330307085660, −11.83038449638971, −11.63179172021720, −11.01233359512141, −10.40309173190206, −9.943783101243967, −9.272323330327034, −8.691027183603121, −7.709022955811525, −6.920461067556201, −6.704575706174788, −5.800882700277038, −5.588395368535054, −4.803531647470070, −4.204412590789519, −3.369595035390939, −2.284296148257007, −1.118396845796386, −0.8242952598730138, 0.8242952598730138, 1.118396845796386, 2.284296148257007, 3.369595035390939, 4.204412590789519, 4.803531647470070, 5.588395368535054, 5.800882700277038, 6.704575706174788, 6.920461067556201, 7.709022955811525, 8.691027183603121, 9.272323330327034, 9.943783101243967, 10.40309173190206, 11.01233359512141, 11.63179172021720, 11.83038449638971, 12.64330307085660, 12.94638284275690, 13.77505012694321, 14.26758835949399, 14.95357848854886, 15.65953335370342, 16.24456183848685

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