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Properties

Label 2-2e10-32.13-c1-0-4
Degree $2$
Conductor $1024$
Sign $-0.321 - 0.946i$
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  + (−0.184 + 0.445i)3-s + (−1.36 + 0.565i)5-s + (0.135 − 0.135i)7-s + (1.95 + 1.95i)9-s + (−1.29 − 3.12i)11-s + (−2.29 − 0.951i)13-s − 0.713i·15-s + 3.11i·17-s + (5.99 + 2.48i)19-s + (0.0352 + 0.0851i)21-s + (5.18 + 5.18i)23-s + (−1.98 + 1.98i)25-s + (−2.57 + 1.06i)27-s + (−1.79 + 4.33i)29-s − 7.44·31-s + ⋯
L(s)  = 1  + (−0.106 + 0.257i)3-s + (−0.610 + 0.253i)5-s + (0.0510 − 0.0510i)7-s + (0.652 + 0.652i)9-s + (−0.390 − 0.941i)11-s + (−0.637 − 0.263i)13-s − 0.184i·15-s + 0.754i·17-s + (1.37 + 0.569i)19-s + (0.00769 + 0.0185i)21-s + (1.08 + 1.08i)23-s + (−0.397 + 0.397i)25-s + (−0.494 + 0.204i)27-s + (−0.333 + 0.804i)29-s − 1.33·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.321 - 0.946i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.035630723\)
\(L(\frac12)\) \(\approx\) \(1.035630723\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.184 - 0.445i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.36 - 0.565i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.135 + 0.135i)T - 7iT^{2} \)
11 \( 1 + (1.29 + 3.12i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (2.29 + 0.951i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 3.11iT - 17T^{2} \)
19 \( 1 + (-5.99 - 2.48i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.18 - 5.18i)T + 23iT^{2} \)
29 \( 1 + (1.79 - 4.33i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
37 \( 1 + (8.44 - 3.49i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-4.27 - 4.27i)T + 41iT^{2} \)
43 \( 1 + (1.79 + 4.33i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 12.0iT - 47T^{2} \)
53 \( 1 + (-1.41 - 3.42i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (2.81 - 1.16i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.61 - 8.72i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (2.96 - 7.15i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (2.86 - 2.86i)T - 71iT^{2} \)
73 \( 1 + (-2.49 - 2.49i)T + 73iT^{2} \)
79 \( 1 - 8.39iT - 79T^{2} \)
83 \( 1 + (-13.0 - 5.42i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-4.96 + 4.96i)T - 89iT^{2} \)
97 \( 1 + 2.87T + 97T^{2} \)
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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

−10.32420818841171608243572971881, −9.439997291622352202860845377485, −8.482452287982323209219961066957, −7.46577199275049320599137298451, −7.24386230753493762660256397346, −5.61795497608295098982669049569, −5.17124432109056134388507428998, −3.82677723362167323250010110902, −3.15310236766790804126018763344, −1.51274201244891721638003791822, 0.49304778112501517418845998470, 2.07651230471672866108559928635, 3.37967550417031385514706066626, 4.53708318059892395379970747204, 5.13648253894780819452684989426, 6.49885405608618332074730501290, 7.37932470158026034673436083232, 7.66036654264153103094511607109, 9.136554429981925408224406450472, 9.478774607708029890597274924944

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