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Properties

Label 2-1280-40.19-c2-0-51
Degree $2$
Conductor $1280$
Sign $0.998 + 0.0505i$
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  + 5.46i·3-s + (−3.70 + 3.35i)5-s − 5.17·7-s − 20.8·9-s + 3.02·11-s + 19.8·13-s + (−18.3 − 20.2i)15-s − 29.5i·17-s − 19.7·19-s − 28.2i·21-s + 6.31·23-s + (2.52 − 24.8i)25-s − 64.7i·27-s − 19.8i·29-s + 19.3i·31-s + ⋯
L(s)  = 1  + 1.82i·3-s + (−0.741 + 0.670i)5-s − 0.738·7-s − 2.31·9-s + 0.275·11-s + 1.52·13-s + (−1.22 − 1.35i)15-s − 1.73i·17-s − 1.03·19-s − 1.34i·21-s + 0.274·23-s + (0.100 − 0.994i)25-s − 2.39i·27-s − 0.683i·29-s + 0.623i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0505i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.998 + 0.0505i$
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: \((2,\ 1280,\ (\ :1),\ 0.998 + 0.0505i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6961623354\)
\(L(\frac12)\) \(\approx\) \(0.6961623354\)
\(L(2)\) 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 \( 1 + (3.70 - 3.35i)T \)
good3 \( 1 - 5.46iT - 9T^{2} \)
7 \( 1 + 5.17T + 49T^{2} \)
11 \( 1 - 3.02T + 121T^{2} \)
13 \( 1 - 19.8T + 169T^{2} \)
17 \( 1 + 29.5iT - 289T^{2} \)
19 \( 1 + 19.7T + 361T^{2} \)
23 \( 1 - 6.31T + 529T^{2} \)
29 \( 1 + 19.8iT - 841T^{2} \)
31 \( 1 - 19.3iT - 961T^{2} \)
37 \( 1 - 7.92T + 1.36e3T^{2} \)
41 \( 1 + 66.5T + 1.68e3T^{2} \)
43 \( 1 + 8.67iT - 1.84e3T^{2} \)
47 \( 1 - 87.7T + 2.20e3T^{2} \)
53 \( 1 - 5.66T + 2.80e3T^{2} \)
59 \( 1 + 62.1T + 3.48e3T^{2} \)
61 \( 1 + 25.2iT - 3.72e3T^{2} \)
67 \( 1 + 52.6iT - 4.48e3T^{2} \)
71 \( 1 + 90.9iT - 5.04e3T^{2} \)
73 \( 1 - 11.9iT - 5.32e3T^{2} \)
79 \( 1 - 91.8iT - 6.24e3T^{2} \)
83 \( 1 - 42.9iT - 6.88e3T^{2} \)
89 \( 1 - 48.1T + 7.92e3T^{2} \)
97 \( 1 - 5.13iT - 9.40e3T^{2} \)
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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

−9.469284242578036671015575868393, −8.917194445256501657474674945065, −8.109277659319141479693381943732, −6.83197048317367327418064659191, −6.11597282663123925327371095712, −5.01809222941731065883369724736, −4.11949474507580494296960489308, −3.49266911299750929193224692006, −2.77908668592585363485536405391, −0.24337320818279443779442667557, 1.00539210389008356133169752360, 1.81973613841998427072539853473, 3.26940284506839605428887991133, 4.12131300866225522509905073263, 5.72559243033973564454436938498, 6.29036169629307194981954844828, 6.96515357647423910676191657834, 7.935281762106498115430201612375, 8.544680064416466522594776698902, 8.950772457568073043045835334438

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