Properties

Label 2-1280-8.5-c1-0-9
Degree $2$
Conductor $1280$
Sign $0.707 - 0.707i$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s − 4·7-s + 3·9-s − 4i·11-s + 2i·13-s + 2·17-s + 4i·19-s + 4·23-s − 25-s + 2i·29-s + 8·31-s − 4i·35-s + 6i·37-s + 6·41-s + 8i·43-s + ⋯
L(s)  = 1  + 0.447i·5-s − 1.51·7-s + 9-s − 1.20i·11-s + 0.554i·13-s + 0.485·17-s + 0.917i·19-s + 0.834·23-s − 0.200·25-s + 0.371i·29-s + 1.43·31-s − 0.676i·35-s + 0.986i·37-s + 0.937·41-s + 1.21i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.427581995\)
\(L(\frac12)\) \(\approx\) \(1.427581995\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 - 3T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.866639121820695831092662345320, −9.118613976944968813447542278707, −8.126689057602863281207603802207, −7.16963968563535773738178697267, −6.42643138239113102869084538153, −5.90031244472374950454902684352, −4.51059440765599261398820940446, −3.48431216676264882025228505285, −2.85511336048940103964150390171, −1.12674084242442291587603208345, 0.72994933058938776222132328125, 2.30841819569299581978980749726, 3.43963126844855277340650502455, 4.42047506778467486043921333025, 5.26677198803139651019079527498, 6.44690641588573265126419483500, 7.02739250018173297176270594124, 7.81045122553104118177974239225, 8.987203995116236834822125741331, 9.735294771613275697138642101526

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