Properties

Label 2-1280-1.1-c1-0-10
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  + 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: $\chi_{1280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.793241296\)
\(L(\frac12)\) \(\approx\) \(1.793241296\)
\(L(\frac{3}{2})\) 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

−9.519403402484082955659663881526, −8.744804945927095622751562737199, −7.994110742227835553764499171902, −7.57468181088586616254724178228, −6.29159702188483236389418914929, −5.42070708329713475773594404896, −4.57308852427161751172368002000, −3.30890989451535714487564718137, −2.67985413200864269839250476138, −1.00700972986514381276153319624, 1.00700972986514381276153319624, 2.67985413200864269839250476138, 3.30890989451535714487564718137, 4.57308852427161751172368002000, 5.42070708329713475773594404896, 6.29159702188483236389418914929, 7.57468181088586616254724178228, 7.994110742227835553764499171902, 8.744804945927095622751562737199, 9.519403402484082955659663881526

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