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Properties

Label 2-1280-40.19-c2-0-64
Degree $2$
Conductor $1280$
Sign $0.458 + 0.888i$
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  + 4.20i·3-s + (−1.52 + 4.76i)5-s + 3.29·7-s − 8.67·9-s − 15.0·11-s − 7.02·13-s + (−20.0 − 6.39i)15-s − 15.9i·17-s + 12.8·19-s + 13.8i·21-s + 3.49·23-s + (−20.3 − 14.4i)25-s + 1.36i·27-s − 13.1i·29-s − 38.2i·31-s + ⋯
L(s)  = 1  + 1.40i·3-s + (−0.304 + 0.952i)5-s + 0.471·7-s − 0.963·9-s − 1.36·11-s − 0.540·13-s + (−1.33 − 0.426i)15-s − 0.936i·17-s + 0.678·19-s + 0.660i·21-s + 0.151·23-s + (−0.814 − 0.579i)25-s + 0.0505i·27-s − 0.453i·29-s − 1.23i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.458 + 0.888i$
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: \((2,\ 1280,\ (\ :1),\ 0.458 + 0.888i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2044525926\)
\(L(\frac12)\) \(\approx\) \(0.2044525926\)
\(L(2)\) 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 \( 1 + (1.52 - 4.76i)T \)
good3 \( 1 - 4.20iT - 9T^{2} \)
7 \( 1 - 3.29T + 49T^{2} \)
11 \( 1 + 15.0T + 121T^{2} \)
13 \( 1 + 7.02T + 169T^{2} \)
17 \( 1 + 15.9iT - 289T^{2} \)
19 \( 1 - 12.8T + 361T^{2} \)
23 \( 1 - 3.49T + 529T^{2} \)
29 \( 1 + 13.1iT - 841T^{2} \)
31 \( 1 + 38.2iT - 961T^{2} \)
37 \( 1 + 68.5T + 1.36e3T^{2} \)
41 \( 1 - 57.3T + 1.68e3T^{2} \)
43 \( 1 + 51.9iT - 1.84e3T^{2} \)
47 \( 1 + 34.0T + 2.20e3T^{2} \)
53 \( 1 + 79.9T + 2.80e3T^{2} \)
59 \( 1 - 114.T + 3.48e3T^{2} \)
61 \( 1 - 115. iT - 3.72e3T^{2} \)
67 \( 1 - 63.3iT - 4.48e3T^{2} \)
71 \( 1 + 40.1iT - 5.04e3T^{2} \)
73 \( 1 + 2.16iT - 5.32e3T^{2} \)
79 \( 1 - 78.1iT - 6.24e3T^{2} \)
83 \( 1 + 61.5iT - 6.88e3T^{2} \)
89 \( 1 - 6.04T + 7.92e3T^{2} \)
97 \( 1 - 135. iT - 9.40e3T^{2} \)
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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

−9.732988589041940339177019729170, −8.592453139524789248345961559003, −7.66020923883105790488635177874, −7.08529442496884701995508038291, −5.65337494658925948374818376641, −5.04634865585581355536874288295, −4.17467401908873598657286100429, −3.18492919873147755885333840875, −2.39935314503492692813319747718, −0.06062386876375944020428884413, 1.20920030334172145406670249759, 2.04494760630203313964073327801, 3.31216788238253748839596073383, 4.81905642085399116388633373992, 5.32121910145322337930294572527, 6.43585930751330345217559818394, 7.37813559160840464662767137765, 7.998621191521279693012177594894, 8.401106984392765842286259605462, 9.484096005994367616409759863438

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