Properties

Label 2-2e10-16.13-c1-0-1
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: $\chi_{1024} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8673514019\)
\(L(\frac12)\) \(\approx\) \(0.8673514019\)
\(L(\frac{3}{2})\) 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

−10.04782725196154851674811644823, −9.336576107691955673425879426308, −8.284663865712695698311853242861, −7.53161215451432126812972597166, −6.70560504915944584189624283534, −5.83792865444966772395299209985, −5.23290258186996399404548335291, −3.57336545675302032102460671821, −2.92830861523978133311758417872, −1.21384734293322764981441273725, 0.48420951722120543289628319774, 2.17830728129228609676492663249, 4.00658069465890824964854811837, 4.51724982003431956238242800376, 5.01694568048205713192358262416, 6.63376834381554036946358940904, 7.13064758306052814034870161589, 8.099930836136042593427402510072, 9.117810221266076677761957087555, 9.899591242853628464837443665601

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