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Properties

Label 2-2e10-16.11-c0-0-1
Degree $2$
Conductor $1024$
Sign $0.923 + 0.382i$
Analytic cond. $0.511042$
Root an. cond. $0.714872$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·9-s + 2·17-s − i·25-s + 2i·41-s − 49-s − 2i·73-s − 81-s + 2i·89-s − 2·97-s − 2·113-s + ⋯
L(s)  = 1  − i·9-s + 2·17-s − i·25-s + 2i·41-s − 49-s − 2i·73-s − 81-s + 2i·89-s − 2·97-s − 2·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.511042\)
Root analytic conductor: \(0.714872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.058607701\)
\(L(\frac12)\) \(\approx\) \(1.058607701\)
\(L(1)\) 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 \( 1 + iT^{2} \)
5 \( 1 + iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - 2T + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.893910092268668263330572347964, −9.488261204488457759579635797779, −8.346355441537798252525637749012, −7.70789545925803540298919383554, −6.60859408187235928946968283513, −5.92874247113194991397927490048, −4.88809645033302291911155758513, −3.74344206426618295783986383647, −2.89973892996504792823034825082, −1.21635891192020931821424454183, 1.56120765405210381840076063051, 2.90119293802153984877451467998, 3.94079953788176670635945800316, 5.20855847154604337766887227419, 5.67301879772382865389720899287, 7.02024563791096199639486316907, 7.70705826865139337364957513415, 8.428557714978460620296815693237, 9.479867020153270945694236091029, 10.20553190313713890790546258311

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