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Properties

Label 2-2e10-16.5-c1-0-9
Degree $2$
Conductor $1024$
Sign $-0.382 - 0.923i$
Analytic cond. $8.17668$
Root an. cond. $2.85948$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)3-s + (1.41 + 1.41i)5-s + 2.82i·7-s + i·9-s + (3 + 3i)11-s + (4.24 − 4.24i)13-s − 2.82·15-s + (3 − 3i)19-s + (−2.82 − 2.82i)21-s + 8.48i·23-s − 0.999i·25-s + (−4 − 4i)27-s + (−1.41 + 1.41i)29-s − 5.65·31-s − 6·33-s + ⋯
L(s)  = 1  + (−0.577 + 0.577i)3-s + (0.632 + 0.632i)5-s + 1.06i·7-s + 0.333i·9-s + (0.904 + 0.904i)11-s + (1.17 − 1.17i)13-s − 0.730·15-s + (0.688 − 0.688i)19-s + (−0.617 − 0.617i)21-s + 1.76i·23-s − 0.199i·25-s + (−0.769 − 0.769i)27-s + (−0.262 + 0.262i)29-s − 1.01·31-s − 1.04·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.568540288\)
\(L(\frac12)\) \(\approx\) \(1.568540288\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (1 - i)T - 3iT^{2} \)
5 \( 1 + (-1.41 - 1.41i)T + 5iT^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 + (-3 - 3i)T + 11iT^{2} \)
13 \( 1 + (-4.24 + 4.24i)T - 13iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-3 + 3i)T - 19iT^{2} \)
23 \( 1 - 8.48iT - 23T^{2} \)
29 \( 1 + (1.41 - 1.41i)T - 29iT^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (-4.24 - 4.24i)T + 37iT^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-1.41 - 1.41i)T + 53iT^{2} \)
59 \( 1 + (1 + i)T + 59iT^{2} \)
61 \( 1 + (4.24 - 4.24i)T - 61iT^{2} \)
67 \( 1 + (-9 + 9i)T - 67iT^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 + 5.65T + 79T^{2} \)
83 \( 1 + (3 - 3i)T - 83iT^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.13288656525932453757064980624, −9.558069830200525598141497124337, −8.755250003896631076642161356469, −7.64319253938600285894260596407, −6.67707227333713701272575466308, −5.57088252483637738116436209740, −5.45636271510916020479813888467, −4.01462721425882279248544942683, −2.90369493158231005466479429586, −1.68817088915288885029569801554, 0.862744730920471668379930644098, 1.58561885084444240047906352936, 3.56690237118909899099067929649, 4.29291193644150157724098194859, 5.63661904033346718017121378947, 6.34588595196539007447578118127, 6.86064128453798018634404726128, 8.026513771407107260464457229679, 9.021305154013056525418343672217, 9.482332232424680750944915045593

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