Properties

Label 2-15680-1.1-c1-0-25
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 + 11-s + 3·13-s − 15-s + 7·17-s − 4·19-s + 25-s − 5·27-s + 5·29-s − 10·31-s + 33-s + 4·37-s + 3·39-s + 10·41-s + 8·43-s + 2·45-s − 47-s + 7·51-s + 4·53-s − 55-s − 4·57-s − 10·61-s − 3·65-s − 12·67-s + 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.301·11-s + 0.832·13-s − 0.258·15-s + 1.69·17-s − 0.917·19-s + 1/5·25-s − 0.962·27-s + 0.928·29-s − 1.79·31-s + 0.174·33-s + 0.657·37-s + 0.480·39-s + 1.56·41-s + 1.21·43-s + 0.298·45-s − 0.145·47-s + 0.980·51-s + 0.549·53-s − 0.134·55-s − 0.529·57-s − 1.28·61-s − 0.372·65-s − 1.46·67-s + 1.42·71-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

\(L(1)\) \(\approx\) \(2.396287895\)
\(L(\frac12)\) \(\approx\) \(2.396287895\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−16.04975384518113, −15.32108482779177, −14.78119796946039, −14.35476627062857, −13.98283415502854, −13.23930831006603, −12.56092227955769, −12.21571024946390, −11.44202151029161, −10.91904839629587, −10.47853584301484, −9.528679188827643, −9.119736403114866, −8.523248856603764, −7.908767293880312, −7.550895917972824, −6.682111773863230, −5.823782708210656, −5.618838612042791, −4.442912738246964, −3.885520116199243, −3.232861607498322, −2.613935572168654, −1.605346064879869, −0.6645532583818589, 0.6645532583818589, 1.605346064879869, 2.613935572168654, 3.232861607498322, 3.885520116199243, 4.442912738246964, 5.618838612042791, 5.823782708210656, 6.682111773863230, 7.550895917972824, 7.908767293880312, 8.523248856603764, 9.119736403114866, 9.528679188827643, 10.47853584301484, 10.91904839629587, 11.44202151029161, 12.21571024946390, 12.56092227955769, 13.23930831006603, 13.98283415502854, 14.35476627062857, 14.78119796946039, 15.32108482779177, 16.04975384518113

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