Properties

Label 2-1280-1.1-c1-0-23
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.732·3-s − 5-s − 1.26·7-s − 2.46·9-s + 3.46·11-s + 3.46·13-s + 0.732·15-s + 3.46·17-s − 2·19-s + 0.928·21-s − 8.19·23-s + 25-s + 4·27-s − 9.46·31-s − 2.53·33-s + 1.26·35-s + 6·37-s − 2.53·39-s − 2.53·41-s − 10.1·43-s + 2.46·45-s − 8.19·47-s − 5.39·49-s − 2.53·51-s − 10.3·53-s − 3.46·55-s + 1.46·57-s + ⋯
L(s)  = 1  − 0.422·3-s − 0.447·5-s − 0.479·7-s − 0.821·9-s + 1.04·11-s + 0.960·13-s + 0.189·15-s + 0.840·17-s − 0.458·19-s + 0.202·21-s − 1.70·23-s + 0.200·25-s + 0.769·27-s − 1.69·31-s − 0.441·33-s + 0.214·35-s + 0.986·37-s − 0.406·39-s − 0.396·41-s − 1.55·43-s + 0.367·45-s − 1.19·47-s − 0.770·49-s − 0.355·51-s − 1.42·53-s − 0.467·55-s + 0.193·57-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: \(1\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.732T + 3T^{2} \)
7 \( 1 + 1.26T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 8.19T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 9.46T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 2.53T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + 8.19T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 - 4.39T + 71T^{2} \)
73 \( 1 - 14.3T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 4.73T + 83T^{2} \)
89 \( 1 + 0.928T + 89T^{2} \)
97 \( 1 + 6.39T + 97T^{2} \)
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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

−9.281360417978357017805351356161, −8.405035320388900658397097166454, −7.76558041424764154092022312267, −6.41377536925402254660438488987, −6.20094081640058115772318205601, −5.08331055880141259994210448130, −3.86601702308792731492741157077, −3.27651483520593012719071023419, −1.62671072304389720712087781313, 0, 1.62671072304389720712087781313, 3.27651483520593012719071023419, 3.86601702308792731492741157077, 5.08331055880141259994210448130, 6.20094081640058115772318205601, 6.41377536925402254660438488987, 7.76558041424764154092022312267, 8.405035320388900658397097166454, 9.281360417978357017805351356161

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