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Properties

Label 2-1280-40.19-c2-0-29
Degree $2$
Conductor $1280$
Sign $0.0639 - 0.997i$
Analytic cond. $34.8774$
Root an. cond. $5.90571$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.547i·3-s + (3.30 + 3.75i)5-s − 10.0·7-s + 8.69·9-s + 17.2·11-s − 4.41·13-s + (2.05 − 1.80i)15-s + 27.0i·17-s − 4.82·19-s + 5.50i·21-s − 15.2·23-s + (−3.19 + 24.7i)25-s − 9.69i·27-s − 2.38i·29-s − 38.0i·31-s + ⋯
L(s)  = 1  − 0.182i·3-s + (0.660 + 0.750i)5-s − 1.43·7-s + 0.966·9-s + 1.56·11-s − 0.339·13-s + (0.137 − 0.120i)15-s + 1.58i·17-s − 0.254·19-s + 0.262i·21-s − 0.663·23-s + (−0.127 + 0.991i)25-s − 0.359i·27-s − 0.0821i·29-s − 1.22i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0639 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0639 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.0639 - 0.997i$
Analytic conductor: \(34.8774\)
Root analytic conductor: \(5.90571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1),\ 0.0639 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.829768131\)
\(L(\frac12)\) \(\approx\) \(1.829768131\)
\(L(2)\) 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 \( 1 + (-3.30 - 3.75i)T \)
good3 \( 1 + 0.547iT - 9T^{2} \)
7 \( 1 + 10.0T + 49T^{2} \)
11 \( 1 - 17.2T + 121T^{2} \)
13 \( 1 + 4.41T + 169T^{2} \)
17 \( 1 - 27.0iT - 289T^{2} \)
19 \( 1 + 4.82T + 361T^{2} \)
23 \( 1 + 15.2T + 529T^{2} \)
29 \( 1 + 2.38iT - 841T^{2} \)
31 \( 1 + 38.0iT - 961T^{2} \)
37 \( 1 - 16.5T + 1.36e3T^{2} \)
41 \( 1 - 13.3T + 1.68e3T^{2} \)
43 \( 1 - 59.7iT - 1.84e3T^{2} \)
47 \( 1 - 62.4T + 2.20e3T^{2} \)
53 \( 1 + 71.5T + 2.80e3T^{2} \)
59 \( 1 + 68.8T + 3.48e3T^{2} \)
61 \( 1 - 40.9iT - 3.72e3T^{2} \)
67 \( 1 - 51.0iT - 4.48e3T^{2} \)
71 \( 1 + 40.4iT - 5.04e3T^{2} \)
73 \( 1 - 35.8iT - 5.32e3T^{2} \)
79 \( 1 - 126. iT - 6.24e3T^{2} \)
83 \( 1 - 75.1iT - 6.88e3T^{2} \)
89 \( 1 - 106.T + 7.92e3T^{2} \)
97 \( 1 - 85.4iT - 9.40e3T^{2} \)
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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

−9.629908980209648634118041788230, −9.250062339913383230220695071074, −7.914643553426381883731357891692, −6.97742147062666711976264154505, −6.25128887660514004590916192784, −6.06023628023864738579170945512, −4.22365439671619082628461564887, −3.64972700431387859873247793502, −2.41945800842489072084990860868, −1.32058782344197500614005172044, 0.55544139502132270568153241795, 1.77746726963888051164089044163, 3.13530260843972310627578719396, 4.14121809926537490557902071867, 4.94742299875151339321667448424, 6.10408194501076889409084691389, 6.69825282823809636591205946861, 7.46585113279592966112701620686, 8.954341474498907764645717165664, 9.302318631827747845895605566688

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