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Properties

Label 2-1280-40.19-c2-0-80
Degree $2$
Conductor $1280$
Sign $-0.960 + 0.277i$
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

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

Invariants

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

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.750805863\)
\(L(\frac12)\) \(\approx\) \(1.750805863\)
\(L(2)\) 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

−8.800553417933800921841609540449, −8.018021401088971203006995844384, −7.56057540321660453033438194349, −7.01680587418715268895817718508, −5.76421698099932070476872117522, −4.75829304844687118256736300582, −4.11956295157099949543927428188, −2.39275487553526954333318266251, −1.52369708123271384956333074851, −0.53781719296885123572968224493, 1.49856523721032735789020124298, 2.99345864024422781051123540729, 4.02925016515314817812341877210, 4.69382703310696103550854142621, 5.15020167883621284061815605821, 6.65533582475018414427750499192, 7.59159363705075867964807225789, 8.231735058055669702438623957441, 9.113794744111013292640575791078, 9.868568718211555028800959821127

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