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Properties

Label 2-2e10-32.29-c1-0-16
Degree $2$
Conductor $1024$
Sign $0.997 + 0.0654i$
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  + (0.445 + 0.184i)3-s + (0.565 + 1.36i)5-s + (0.135 − 0.135i)7-s + (−1.95 − 1.95i)9-s + (3.12 − 1.29i)11-s + (0.951 − 2.29i)13-s + 0.713i·15-s + 3.11i·17-s + (2.48 − 5.99i)19-s + (0.0851 − 0.0352i)21-s + (5.18 + 5.18i)23-s + (1.98 − 1.98i)25-s + (−1.06 − 2.57i)27-s + (−4.33 − 1.79i)29-s + 7.44·31-s + ⋯
L(s)  = 1  + (0.257 + 0.106i)3-s + (0.253 + 0.610i)5-s + (0.0510 − 0.0510i)7-s + (−0.652 − 0.652i)9-s + (0.941 − 0.390i)11-s + (0.263 − 0.637i)13-s + 0.184i·15-s + 0.754i·17-s + (0.569 − 1.37i)19-s + (0.0185 − 0.00769i)21-s + (1.08 + 1.08i)23-s + (0.397 − 0.397i)25-s + (−0.204 − 0.494i)27-s + (−0.804 − 0.333i)29-s + 1.33·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.997 + 0.0654i$
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: \((2,\ 1024,\ (\ :1/2),\ 0.997 + 0.0654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.912407189\)
\(L(\frac12)\) \(\approx\) \(1.912407189\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.445 - 0.184i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.565 - 1.36i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.135 + 0.135i)T - 7iT^{2} \)
11 \( 1 + (-3.12 + 1.29i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.951 + 2.29i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 3.11iT - 17T^{2} \)
19 \( 1 + (-2.48 + 5.99i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-5.18 - 5.18i)T + 23iT^{2} \)
29 \( 1 + (4.33 + 1.79i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 7.44T + 31T^{2} \)
37 \( 1 + (-3.49 - 8.44i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.27 + 4.27i)T + 41iT^{2} \)
43 \( 1 + (-4.33 + 1.79i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 12.0iT - 47T^{2} \)
53 \( 1 + (-3.42 + 1.41i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (1.16 + 2.81i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (8.72 + 3.61i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-7.15 - 2.96i)T + (47.3 + 47.3i)T^{2} \)
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 \( 1 + (-5.42 + 13.0i)T + (-58.6 - 58.6i)T^{2} \)
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

−9.794422661932264723194770953190, −9.136485266166524187615637335098, −8.425657529014114476214654009710, −7.38725563319728393817953415188, −6.43780620576017488869371272794, −5.87885854879569670759335019070, −4.62093308137282677597350148506, −3.38649735709831938319222237826, −2.81859701313070741843705012354, −1.06350014733575390134787636616, 1.25154402252117039907274634508, 2.42466371049021364821495857790, 3.70049900002636326280620304271, 4.78686934110454947242646168848, 5.55534506363633473805876613407, 6.61841802050371427296353766958, 7.47638657825099703296712202331, 8.489944878430947248326976647063, 9.031951854360269780789467808835, 9.765773231563253955861508626640

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