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Properties

Label 2-1280-8.3-c2-0-17
Degree $2$
Conductor $1280$
Sign $-0.707 - 0.707i$
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  + 3.80·3-s + 2.23i·5-s + 8.50i·7-s + 5.47·9-s − 1.79·11-s − 0.472i·13-s + 8.50i·15-s − 23.8·17-s − 9.40·19-s + 32.3i·21-s + 16.1i·23-s − 5.00·25-s − 13.4·27-s − 6.94i·29-s + 47.4i·31-s + ⋯
L(s)  = 1  + 1.26·3-s + 0.447i·5-s + 1.21i·7-s + 0.608·9-s − 0.163·11-s − 0.0363i·13-s + 0.567i·15-s − 1.40·17-s − 0.494·19-s + 1.54i·21-s + 0.700i·23-s − 0.200·25-s − 0.497·27-s − 0.239i·29-s + 1.53i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(34.8774\)
Root analytic conductor: \(5.90571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.909150740\)
\(L(\frac12)\) \(\approx\) \(1.909150740\)
\(L(2)\) 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 \( 1 - 2.23iT \)
good3 \( 1 - 3.80T + 9T^{2} \)
7 \( 1 - 8.50iT - 49T^{2} \)
11 \( 1 + 1.79T + 121T^{2} \)
13 \( 1 + 0.472iT - 169T^{2} \)
17 \( 1 + 23.8T + 289T^{2} \)
19 \( 1 + 9.40T + 361T^{2} \)
23 \( 1 - 16.1iT - 529T^{2} \)
29 \( 1 + 6.94iT - 841T^{2} \)
31 \( 1 - 47.4iT - 961T^{2} \)
37 \( 1 - 26.3iT - 1.36e3T^{2} \)
41 \( 1 - 41.4T + 1.68e3T^{2} \)
43 \( 1 - 2.00T + 1.84e3T^{2} \)
47 \( 1 + 35.3iT - 2.20e3T^{2} \)
53 \( 1 + 21.6iT - 2.80e3T^{2} \)
59 \( 1 + 73.8T + 3.48e3T^{2} \)
61 \( 1 - 26.1iT - 3.72e3T^{2} \)
67 \( 1 - 88.8T + 4.48e3T^{2} \)
71 \( 1 - 39.4iT - 5.04e3T^{2} \)
73 \( 1 + 137.T + 5.32e3T^{2} \)
79 \( 1 + 113. iT - 6.24e3T^{2} \)
83 \( 1 - 21.2T + 6.88e3T^{2} \)
89 \( 1 + 67.4T + 7.92e3T^{2} \)
97 \( 1 + 39.1T + 9.40e3T^{2} \)
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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

−9.502152503252914622471522143628, −8.836678070942861593656918582436, −8.432288074081293304093289819627, −7.47794515295246965407551217506, −6.56897317105693200936076003356, −5.66008632729104372694053686946, −4.54642062689183321948655029992, −3.39445576756201181337833047724, −2.61348083326064778858670369020, −1.90456168102444421342172914644, 0.41809923654717290356479194978, 1.90795510445816607268416886390, 2.84914290736239458010771211549, 4.11286696181439445094872400203, 4.40480350538477399313455528400, 5.90943233583833475794068590968, 6.95269812316936743056522952803, 7.69953226916822218456162875064, 8.379807139878851373595561819414, 9.113398322480063835602541521494

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