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Properties

Label 2-2e10-32.21-c1-0-26
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: $\chi_{1024} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.555 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.902854788\)
\(L(\frac12)\) \(\approx\) \(2.902854788\)
\(L(\frac{3}{2})\) 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

−9.515128357378647684425813902275, −8.671171846975688517221263003181, −8.504940392073052169819065744905, −7.56099461491612359948774426852, −6.52776711806075543197751843743, −5.41587007557406553012687289547, −4.53351162401791338886630748291, −3.33792350013455671338456234742, −2.12391371519517638549590157848, −1.35805152304010573177153629158, 1.81757155596703620603776140719, 2.77041318113722281087099436855, 3.78108105660411370772259424737, 4.41837108578928321435413902816, 5.91471649268991468152163008655, 6.98835899131806116466235033982, 7.44213357003306858836956982069, 8.684626345937724718875174786876, 9.243950895735346279144037373866, 9.877964432209278780843118369868

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