Properties

Label 2-15680-1.1-c1-0-33
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-s + 5-s − 2·9-s + 6·11-s + 4·13-s − 15-s − 2·19-s − 3·23-s + 25-s + 5·27-s + 3·29-s + 8·31-s − 6·33-s + 4·37-s − 4·39-s + 9·41-s + 7·43-s − 2·45-s + 6·53-s + 6·55-s + 2·57-s + 6·59-s − 5·61-s + 4·65-s − 5·67-s + 3·69-s − 6·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 2/3·9-s + 1.80·11-s + 1.10·13-s − 0.258·15-s − 0.458·19-s − 0.625·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s + 1.43·31-s − 1.04·33-s + 0.657·37-s − 0.640·39-s + 1.40·41-s + 1.06·43-s − 0.298·45-s + 0.824·53-s + 0.809·55-s + 0.264·57-s + 0.781·59-s − 0.640·61-s + 0.496·65-s − 0.610·67-s + 0.361·69-s − 0.712·71-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

\(L(1)\) \(\approx\) \(2.390462791\)
\(L(\frac12)\) \(\approx\) \(2.390462791\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
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

−16.21753853403966, −15.50562878047258, −14.71011250940203, −14.33208492004896, −13.84060209008204, −13.31145871684950, −12.50028597564886, −11.98013920524511, −11.52760192683532, −11.03418164050554, −10.40403166601580, −9.771975036373285, −9.015332271813418, −8.709084390013228, −8.052481621487057, −7.082320015123705, −6.437423294044670, −5.985338406813193, −5.716870733739355, −4.419323414476325, −4.234523378709341, −3.223295325651998, −2.445635635982235, −1.380344720946920, −0.7732167722797834, 0.7732167722797834, 1.380344720946920, 2.445635635982235, 3.223295325651998, 4.234523378709341, 4.419323414476325, 5.716870733739355, 5.985338406813193, 6.437423294044670, 7.082320015123705, 8.052481621487057, 8.709084390013228, 9.015332271813418, 9.771975036373285, 10.40403166601580, 11.03418164050554, 11.52760192683532, 11.98013920524511, 12.50028597564886, 13.31145871684950, 13.84060209008204, 14.33208492004896, 14.71011250940203, 15.50562878047258, 16.21753853403966

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