Properties

Label 2-1280-1.1-c1-0-19
Degree $2$
Conductor $1280$
Sign $1$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s + 5-s + 4.24·7-s − 0.999·9-s + 5.65·11-s − 2·13-s + 1.41·15-s + 6·17-s − 2.82·19-s + 6·21-s − 7.07·23-s + 25-s − 5.65·27-s − 4·29-s − 2.82·31-s + 8.00·33-s + 4.24·35-s + 2·37-s − 2.82·39-s + 8·41-s − 1.41·43-s − 0.999·45-s − 1.41·47-s + 10.9·49-s + 8.48·51-s − 2·53-s + 5.65·55-s + ⋯
L(s)  = 1  + 0.816·3-s + 0.447·5-s + 1.60·7-s − 0.333·9-s + 1.70·11-s − 0.554·13-s + 0.365·15-s + 1.45·17-s − 0.648·19-s + 1.30·21-s − 1.47·23-s + 0.200·25-s − 1.08·27-s − 0.742·29-s − 0.508·31-s + 1.39·33-s + 0.717·35-s + 0.328·37-s − 0.452·39-s + 1.24·41-s − 0.215·43-s − 0.149·45-s − 0.206·47-s + 1.57·49-s + 1.18·51-s − 0.274·53-s + 0.762·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $1$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.866495792\)
\(L(\frac12)\) \(\approx\) \(2.866495792\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 - 1.41T + 3T^{2} \)
7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 7.07T + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 + 1.41T + 43T^{2} \)
47 \( 1 + 1.41T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 4.24T + 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−9.417813960759149650388470402767, −8.934543866025143240598883397938, −7.965800338953623128029847401656, −7.60923321559064389382275585544, −6.26415304508219311984899254644, −5.49936600934448183561518176378, −4.40058944195331033734735040382, −3.58082100235383699110499102578, −2.22259305792153845108702480225, −1.45358908317161389346517381853, 1.45358908317161389346517381853, 2.22259305792153845108702480225, 3.58082100235383699110499102578, 4.40058944195331033734735040382, 5.49936600934448183561518176378, 6.26415304508219311984899254644, 7.60923321559064389382275585544, 7.965800338953623128029847401656, 8.934543866025143240598883397938, 9.417813960759149650388470402767

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