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Properties

Label 2-2e10-16.5-c1-0-8
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  + (−2.41 + 2.41i)3-s − 8.65i·9-s + (1.58 + 1.58i)11-s + 5.65·17-s + (3.24 − 3.24i)19-s − 5i·25-s + (13.6 + 13.6i)27-s − 7.65·33-s + 6i·41-s + (0.757 + 0.757i)43-s + 7·49-s + (−13.6 + 13.6i)51-s + 15.6i·57-s + (−4.07 − 4.07i)59-s + (−11.2 + 11.2i)67-s + ⋯
L(s)  = 1  + (−1.39 + 1.39i)3-s − 2.88i·9-s + (0.478 + 0.478i)11-s + 1.37·17-s + (0.743 − 0.743i)19-s − i·25-s + (2.62 + 2.62i)27-s − 1.33·33-s + 0.937i·41-s + (0.115 + 0.115i)43-s + 49-s + (−1.91 + 1.91i)51-s + 2.07i·57-s + (−0.530 − 0.530i)59-s + (−1.37 + 1.37i)67-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

\(L(1)\) \(\approx\) \(1.010396126\)
\(L(\frac12)\) \(\approx\) \(1.010396126\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 - 2.41i)T - 3iT^{2} \)
5 \( 1 + 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (-1.58 - 1.58i)T + 11iT^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
19 \( 1 + (-3.24 + 3.24i)T - 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (-0.757 - 0.757i)T + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + (4.07 + 4.07i)T + 59iT^{2} \)
61 \( 1 - 61iT^{2} \)
67 \( 1 + (11.2 - 11.2i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 16.9iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-10.4 + 10.4i)T - 83iT^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−10.08004176546427987616874422179, −9.622935792241844870686382966508, −8.778919431737192374071267914984, −7.40952164660178611328538749711, −6.43565971960738899555893431919, −5.67956271391557827951264486657, −4.88488646434959658482900002188, −4.14120439923395761381357450727, −3.13879645580351784091299379621, −0.905955734849390577489852275080, 0.836159346526401932713298269125, 1.79760647417396386710121439611, 3.38248833937576333487257607125, 4.93597989329625409920712744339, 5.74911921386079266111980617673, 6.21516259363083371826697930812, 7.42944809034766431319288188032, 7.60854769674339296289726982861, 8.822616116775596346644113465622, 10.05625177016000565181842872605

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