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Properties

Label 2-1280-40.19-c2-0-50
Degree $2$
Conductor $1280$
Sign $0.277 + 0.960i$
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  − 4.06i·3-s + (4.37 + 2.41i)5-s − 13.1·7-s − 7.55·9-s + 11.8·11-s + 18.9·13-s + (9.82 − 17.8i)15-s − 5.69i·17-s + 4.21·19-s + 53.5i·21-s + 3.23·23-s + (13.3 + 21.1i)25-s − 5.86i·27-s + 25.9i·29-s + 42.1i·31-s + ⋯
L(s)  = 1  − 1.35i·3-s + (0.875 + 0.482i)5-s − 1.88·7-s − 0.839·9-s + 1.07·11-s + 1.46·13-s + (0.654 − 1.18i)15-s − 0.334i·17-s + 0.221·19-s + 2.55i·21-s + 0.140·23-s + (0.533 + 0.845i)25-s − 0.217i·27-s + 0.895i·29-s + 1.35i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.277 + 0.960i$
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: \((2,\ 1280,\ (\ :1),\ 0.277 + 0.960i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.127404980\)
\(L(\frac12)\) \(\approx\) \(2.127404980\)
\(L(2)\) 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 \( 1 + (-4.37 - 2.41i)T \)
good3 \( 1 + 4.06iT - 9T^{2} \)
7 \( 1 + 13.1T + 49T^{2} \)
11 \( 1 - 11.8T + 121T^{2} \)
13 \( 1 - 18.9T + 169T^{2} \)
17 \( 1 + 5.69iT - 289T^{2} \)
19 \( 1 - 4.21T + 361T^{2} \)
23 \( 1 - 3.23T + 529T^{2} \)
29 \( 1 - 25.9iT - 841T^{2} \)
31 \( 1 - 42.1iT - 961T^{2} \)
37 \( 1 + 12.4T + 1.36e3T^{2} \)
41 \( 1 - 16.7T + 1.68e3T^{2} \)
43 \( 1 + 71.8iT - 1.84e3T^{2} \)
47 \( 1 - 18.2T + 2.20e3T^{2} \)
53 \( 1 - 92.6T + 2.80e3T^{2} \)
59 \( 1 - 17.2T + 3.48e3T^{2} \)
61 \( 1 + 56.4iT - 3.72e3T^{2} \)
67 \( 1 - 31.8iT - 4.48e3T^{2} \)
71 \( 1 + 94.3iT - 5.04e3T^{2} \)
73 \( 1 + 123. iT - 5.32e3T^{2} \)
79 \( 1 + 17.9iT - 6.24e3T^{2} \)
83 \( 1 - 5.00iT - 6.88e3T^{2} \)
89 \( 1 + 90.1T + 7.92e3T^{2} \)
97 \( 1 + 129. iT - 9.40e3T^{2} \)
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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

−9.115752684319649320013180451552, −8.751924369636462800839326754300, −7.20319865278142908810713120100, −6.79223120191124588520182368590, −6.27843022788792288344028022352, −5.59641011781533789424279818298, −3.70092067364480104294351558084, −3.02349483211219944058884966295, −1.80487356037744603830058411944, −0.798170656352583218801592520346, 1.00132328554851634043304810559, 2.71194164151177472688680637763, 3.82104430663917944836223256800, 4.16256565239021067249365556316, 5.66726170465031632892151691353, 6.07100766857347118114616354831, 6.87084596008800059692917047005, 8.495720572357852347492395437081, 9.156890798035164963441778584023, 9.666936832767799727298864677853

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