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Properties

Label 2-2e10-32.29-c1-0-30
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  + (0.969 + 0.401i)3-s + (−1.44 − 3.49i)5-s + (3.21 − 3.21i)7-s + (−1.34 − 1.34i)9-s + (−0.878 + 0.363i)11-s + (−0.985 + 2.37i)13-s − 3.97i·15-s − 1.34i·17-s + (−1.28 + 3.10i)19-s + (4.41 − 1.82i)21-s + (−4.30 − 4.30i)23-s + (−6.60 + 6.60i)25-s + (−1.96 − 4.74i)27-s + (−1.44 − 0.600i)29-s + 3.69·31-s + ⋯
L(s)  = 1  + (0.559 + 0.231i)3-s + (−0.648 − 1.56i)5-s + (1.21 − 1.21i)7-s + (−0.447 − 0.447i)9-s + (−0.264 + 0.109i)11-s + (−0.273 + 0.659i)13-s − 1.02i·15-s − 0.326i·17-s + (−0.295 + 0.712i)19-s + (0.963 − 0.398i)21-s + (−0.896 − 0.896i)23-s + (−1.32 + 1.32i)25-s + (−0.378 − 0.914i)27-s + (−0.269 − 0.111i)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

\(L(1)\) \(\approx\) \(1.486689191\)
\(L(\frac12)\) \(\approx\) \(1.486689191\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.969 - 0.401i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (1.44 + 3.49i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-3.21 + 3.21i)T - 7iT^{2} \)
11 \( 1 + (0.878 - 0.363i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (0.985 - 2.37i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 1.34iT - 17T^{2} \)
19 \( 1 + (1.28 - 3.10i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.30 + 4.30i)T + 23iT^{2} \)
29 \( 1 + (1.44 + 0.600i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 3.69T + 31T^{2} \)
37 \( 1 + (-1.67 - 4.03i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-5.34 - 5.34i)T + 41iT^{2} \)
43 \( 1 + (7.27 - 3.01i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 1.02iT - 47T^{2} \)
53 \( 1 + (-6.49 + 2.69i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.44 + 5.90i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (4.42 + 1.83i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-6.23 - 2.58i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-5.75 + 5.75i)T - 71iT^{2} \)
73 \( 1 + (-2.94 - 2.94i)T + 73iT^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + (-4.66 + 11.2i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-9.04 + 9.04i)T - 89iT^{2} \)
97 \( 1 - 8.41T + 97T^{2} \)
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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

−9.555723792824970275310083091199, −8.605148701867859461455627148991, −8.139103234816455692640579823069, −7.56686784717945107278862634215, −6.22843662192720009624505786119, −4.81293869743035650789676371723, −4.49371849316602528382395337044, −3.61939777817779024449235791007, −1.88044852477070758526693772519, −0.61637494841041813255907377169, 2.19637285231286192998620895428, 2.67225607350688221063880687245, 3.77952754279534003978100184389, 5.16832004719152453109329402699, 5.92057835045265707074517173321, 7.11836171895680213916977952916, 7.913248526595324150340128038287, 8.244401835761192704076802503741, 9.254592563134713781686678990707, 10.48650454578758967466809291940

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