Properties

Label 2-15680-1.1-c1-0-48
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 − 11-s − 13-s − 3·15-s + 3·17-s + 8·19-s + 4·23-s + 25-s + 9·27-s − 3·29-s + 6·31-s − 3·33-s + 8·37-s − 3·39-s − 10·41-s + 12·43-s − 6·45-s − 3·47-s + 9·51-s − 12·53-s + 55-s + 24·57-s + 2·61-s + 65-s − 4·67-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s + 2·9-s − 0.301·11-s − 0.277·13-s − 0.774·15-s + 0.727·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s + 1.73·27-s − 0.557·29-s + 1.07·31-s − 0.522·33-s + 1.31·37-s − 0.480·39-s − 1.56·41-s + 1.82·43-s − 0.894·45-s − 0.437·47-s + 1.26·51-s − 1.64·53-s + 0.134·55-s + 3.17·57-s + 0.256·61-s + 0.124·65-s − 0.488·67-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

\(L(1)\) \(\approx\) \(4.628070488\)
\(L(\frac12)\) \(\approx\) \(4.628070488\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−15.76606945067305, −15.44423853846818, −14.76480130531480, −14.34975720847417, −13.93075962484608, −13.23040315347590, −12.92845595150916, −12.10538276078292, −11.63192679117833, −10.84985150084215, −10.06771588531276, −9.583624914251186, −9.211377689830413, −8.444673030973758, −7.909507264581539, −7.499532780430823, −7.060550893444952, −6.049703100713662, −5.154302038175277, −4.530441636354457, −3.740200057843772, −2.989633985347454, −2.855925499393895, −1.702071063334869, −0.8920413177809668, 0.8920413177809668, 1.702071063334869, 2.855925499393895, 2.989633985347454, 3.740200057843772, 4.530441636354457, 5.154302038175277, 6.049703100713662, 7.060550893444952, 7.499532780430823, 7.909507264581539, 8.444673030973758, 9.211377689830413, 9.583624914251186, 10.06771588531276, 10.84985150084215, 11.63192679117833, 12.10538276078292, 12.92845595150916, 13.23040315347590, 13.93075962484608, 14.34975720847417, 14.76480130531480, 15.44423853846818, 15.76606945067305

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