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Properties

Label 2-1280-20.7-c1-0-9
Degree $2$
Conductor $1280$
Sign $0.340 + 0.940i$
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  + (−2.18 − 2.18i)3-s + (−2.18 − 0.456i)5-s + (−1.79 + 1.79i)7-s + 6.58i·9-s − 0.913i·11-s + (−1.73 + 1.73i)13-s + (3.79 + 5.79i)15-s + (3 + 3i)17-s + 3.46·19-s + 7.84·21-s + (−3.79 − 3.79i)23-s + (4.58 + 1.99i)25-s + (7.84 − 7.84i)27-s + 5.29i·29-s − 7.58i·31-s + ⋯
L(s)  = 1  + (−1.26 − 1.26i)3-s + (−0.978 − 0.204i)5-s + (−0.677 + 0.677i)7-s + 2.19i·9-s − 0.275i·11-s + (−0.480 + 0.480i)13-s + (0.978 + 1.49i)15-s + (0.727 + 0.727i)17-s + 0.794·19-s + 1.71·21-s + (−0.790 − 0.790i)23-s + (0.916 + 0.399i)25-s + (1.50 − 1.50i)27-s + 0.982i·29-s − 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.340 + 0.940i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.340 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5400692328\)
\(L(\frac12)\) \(\approx\) \(0.5400692328\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 + 0.456i)T \)
good3 \( 1 + (2.18 + 2.18i)T + 3iT^{2} \)
7 \( 1 + (1.79 - 1.79i)T - 7iT^{2} \)
11 \( 1 + 0.913iT - 11T^{2} \)
13 \( 1 + (1.73 - 1.73i)T - 13iT^{2} \)
17 \( 1 + (-3 - 3i)T + 17iT^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + (3.79 + 3.79i)T + 23iT^{2} \)
29 \( 1 - 5.29iT - 29T^{2} \)
31 \( 1 + 7.58iT - 31T^{2} \)
37 \( 1 + (5.19 + 5.19i)T + 37iT^{2} \)
41 \( 1 + 1.58T + 41T^{2} \)
43 \( 1 + (-0.361 - 0.361i)T + 43iT^{2} \)
47 \( 1 + (3.79 - 3.79i)T - 47iT^{2} \)
53 \( 1 + (2.64 - 2.64i)T - 53iT^{2} \)
59 \( 1 + 5.29T + 59T^{2} \)
61 \( 1 - 6.20T + 61T^{2} \)
67 \( 1 + (0.361 - 0.361i)T - 67iT^{2} \)
71 \( 1 - 4.41iT - 71T^{2} \)
73 \( 1 + (-8.58 + 8.58i)T - 73iT^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + (3.10 + 3.10i)T + 83iT^{2} \)
89 \( 1 + 15.1iT - 89T^{2} \)
97 \( 1 + (8.58 + 8.58i)T + 97iT^{2} \)
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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

−9.512522605204359896934084223042, −8.441661877820457721150215333855, −7.69045529844118278695120126039, −7.03473117994460286894866417565, −6.16966281071644743678741039806, −5.57599550695990528519716056814, −4.56871173359601038070173664651, −3.26027568713055316325387413965, −1.88129602235681715378954823969, −0.49688005640538729742901301332, 0.63955474706874377255667080113, 3.27603842700436630257414950209, 3.74366877949609161995185041045, 4.82234340285665190836992479992, 5.35997060140309945846907691050, 6.51748199360778313280708201197, 7.20923417338237605405426066874, 8.129464582387403223141981633296, 9.481943448318870547770568406520, 9.973049862252298856444322372356

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