Properties

Label 2-1280-1.1-c1-0-17
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  + 2.73·3-s + 5-s − 0.732·7-s + 4.46·9-s + 2·11-s − 3.46·13-s + 2.73·15-s + 3.46·17-s + 0.535·19-s − 2·21-s + 6.19·23-s + 25-s + 3.99·27-s + 6.92·29-s − 5.46·31-s + 5.46·33-s − 0.732·35-s + 2·37-s − 9.46·39-s − 1.46·41-s + 5.26·43-s + 4.46·45-s + 3.26·47-s − 6.46·49-s + 9.46·51-s − 11.4·53-s + 2·55-s + ⋯
L(s)  = 1  + 1.57·3-s + 0.447·5-s − 0.276·7-s + 1.48·9-s + 0.603·11-s − 0.960·13-s + 0.705·15-s + 0.840·17-s + 0.122·19-s − 0.436·21-s + 1.29·23-s + 0.200·25-s + 0.769·27-s + 1.28·29-s − 0.981·31-s + 0.951·33-s − 0.123·35-s + 0.328·37-s − 1.51·39-s − 0.228·41-s + 0.803·43-s + 0.665·45-s + 0.476·47-s − 0.923·49-s + 1.32·51-s − 1.57·53-s + 0.269·55-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

\(L(1)\) \(\approx\) \(3.120079751\)
\(L(\frac12)\) \(\approx\) \(3.120079751\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 3T^{2} \)
7 \( 1 + 0.732T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 0.535T + 19T^{2} \)
23 \( 1 - 6.19T + 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 - 5.26T + 43T^{2} \)
47 \( 1 - 3.26T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 - 7.46T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + 7.46T + 73T^{2} \)
79 \( 1 + 1.07T + 79T^{2} \)
83 \( 1 - 1.26T + 83T^{2} \)
89 \( 1 + 8.92T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−9.505787242908214611883146459689, −8.994570710974860012793157397802, −8.119786568776193441008436788120, −7.35522326405815039229326843306, −6.61567147189225814031160257017, −5.38047197900176002481020886461, −4.33615390727861181813506908219, −3.24330185652132838407459964870, −2.63435167007807001565409260723, −1.41539907066851654198387062095, 1.41539907066851654198387062095, 2.63435167007807001565409260723, 3.24330185652132838407459964870, 4.33615390727861181813506908219, 5.38047197900176002481020886461, 6.61567147189225814031160257017, 7.35522326405815039229326843306, 8.119786568776193441008436788120, 8.994570710974860012793157397802, 9.505787242908214611883146459689

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