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Properties

Label 2-2e10-32.29-c1-0-14
Degree $2$
Conductor $1024$
Sign $-0.555 + 0.831i$
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  + (−2.27 − 0.942i)3-s + (1.03 + 2.49i)5-s + (−1.37 + 1.37i)7-s + (2.17 + 2.17i)9-s + (−4.12 + 1.70i)11-s + (−2.01 + 4.86i)13-s − 6.66i·15-s − 6.31i·17-s + (1.95 − 4.72i)19-s + (4.41 − 1.82i)21-s + (0.288 + 0.288i)23-s + (−1.63 + 1.63i)25-s + (−0.0660 − 0.159i)27-s + (1.03 + 0.428i)29-s + 3.69·31-s + ⋯
L(s)  = 1  + (−1.31 − 0.544i)3-s + (0.462 + 1.11i)5-s + (−0.518 + 0.518i)7-s + (0.723 + 0.723i)9-s + (−1.24 + 0.515i)11-s + (−0.558 + 1.34i)13-s − 1.72i·15-s − 1.53i·17-s + (0.449 − 1.08i)19-s + (0.963 − 0.398i)21-s + (0.0602 + 0.0602i)23-s + (−0.327 + 0.327i)25-s + (−0.0127 − 0.0306i)27-s + (0.192 + 0.0796i)29-s + 0.663·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.555 + 0.831i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.555 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2080024255\)
\(L(\frac12)\) \(\approx\) \(0.2080024255\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.27 + 0.942i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.03 - 2.49i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.37 - 1.37i)T - 7iT^{2} \)
11 \( 1 + (4.12 - 1.70i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.01 - 4.86i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 6.31iT - 17T^{2} \)
19 \( 1 + (-1.95 + 4.72i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.288 - 0.288i)T + 23iT^{2} \)
29 \( 1 + (-1.03 - 0.428i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 3.69T + 31T^{2} \)
37 \( 1 + (4.32 + 10.4i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.68 + 1.68i)T + 41iT^{2} \)
43 \( 1 + (4.03 - 1.67i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 4.83iT - 47T^{2} \)
53 \( 1 + (-0.501 + 0.207i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (3.00 + 7.25i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (3.39 + 1.40i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-0.299 - 0.123i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-6.54 + 6.54i)T - 71iT^{2} \)
73 \( 1 + (5.53 + 5.53i)T + 73iT^{2} \)
79 \( 1 + 0.877iT - 79T^{2} \)
83 \( 1 + (-1.41 + 3.42i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (6.46 - 6.46i)T - 89iT^{2} \)
97 \( 1 + 3.58T + 97T^{2} \)
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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

−9.768212435314321270808756358261, −9.120061808502335932309363537161, −7.47883868245619848407442698778, −6.94349974328856543272195653157, −6.44217378521982745149428082057, −5.37753686994415389140853931532, −4.80562950117597947796573756693, −2.93980189049731485887775303037, −2.17777794060970914316579840467, −0.12172823443239820711245956524, 1.17062157658125656632698954452, 3.09963837161911830935543904157, 4.34831867897968868739159805315, 5.33450379448881436112593779305, 5.59329488676165516163853788429, 6.52465556222939007423426383553, 7.945936876627250020428714156336, 8.438426911889247795568273115961, 9.904129315188418259488213648176, 10.20863686451145043330808522664

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