Properties

Label 2-1280-20.3-c1-0-9
Degree $2$
Conductor $1280$
Sign $0.850 - 0.525i$
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  + (1 − i)3-s + (−2 − i)5-s + (−1 − i)7-s + i·9-s + 4i·11-s + (3 + 3i)13-s + (−3 + i)15-s + (−3 + 3i)17-s + 6·19-s − 2·21-s + (−3 + 3i)23-s + (3 + 4i)25-s + (4 + 4i)27-s − 2i·29-s − 6i·31-s + ⋯
L(s)  = 1  + (0.577 − 0.577i)3-s + (−0.894 − 0.447i)5-s + (−0.377 − 0.377i)7-s + 0.333i·9-s + 1.20i·11-s + (0.832 + 0.832i)13-s + (−0.774 + 0.258i)15-s + (−0.727 + 0.727i)17-s + 1.37·19-s − 0.436·21-s + (−0.625 + 0.625i)23-s + (0.600 + 0.800i)25-s + (0.769 + 0.769i)27-s − 0.371i·29-s − 1.07i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (1023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.500711276\)
\(L(\frac12)\) \(\approx\) \(1.500711276\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
good3 \( 1 + (-1 + i)T - 3iT^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + (3 - 3i)T - 23iT^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 6iT - 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 + (-9 - 9i)T + 47iT^{2} \)
53 \( 1 + (-5 - 5i)T + 53iT^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 + (-9 - 9i)T + 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (3 - 3i)T - 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−9.539994810994042521246872505391, −8.873495929082855836628155208453, −7.954351199074866528575922904573, −7.43613424880558753341772018411, −6.76078023017890266240201548927, −5.53429960314262907686507309015, −4.31392515244849746139924828302, −3.81078526827458449596811091803, −2.39747503569600525383894734055, −1.30235978816778227887700498647, 0.65582297781718012055007100399, 2.93550550580576916165475114295, 3.25006326354692888511777464048, 4.16015594184457010061608941458, 5.38203084575206009875689790453, 6.32522873187797143820731578818, 7.16343907583786005929418723647, 8.319963941414979635705062489178, 8.617914218246992681510748451867, 9.520284011433220289967643089439

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