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Properties

Label 2-2e10-32.13-c1-0-31
Degree $2$
Conductor $1024$
Sign $-0.751 - 0.659i$
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  + (1.12 − 2.70i)3-s + (−0.366 + 0.151i)5-s + (−3.06 + 3.06i)7-s + (−3.95 − 3.95i)9-s + (−1.51 − 3.66i)11-s + (−1.88 − 0.780i)13-s + 1.16i·15-s + 4.54i·17-s + (−0.534 − 0.221i)19-s + (4.86 + 11.7i)21-s + (−4.41 − 4.41i)23-s + (−3.42 + 3.42i)25-s + (−7.03 + 2.91i)27-s + (−1.96 + 4.74i)29-s + 0.0539·31-s + ⋯
L(s)  = 1  + (0.647 − 1.56i)3-s + (−0.163 + 0.0678i)5-s + (−1.15 + 1.15i)7-s + (−1.31 − 1.31i)9-s + (−0.457 − 1.10i)11-s + (−0.522 − 0.216i)13-s + 0.299i·15-s + 1.10i·17-s + (−0.122 − 0.0508i)19-s + (1.06 + 2.56i)21-s + (−0.921 − 0.921i)23-s + (−0.684 + 0.684i)25-s + (−1.35 + 0.560i)27-s + (−0.365 + 0.881i)29-s + 0.00969·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.751 - 0.659i$
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: \((2,\ 1024,\ (\ :1/2),\ -0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3344252858\)
\(L(\frac12)\) \(\approx\) \(0.3344252858\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 + 2.70i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.366 - 0.151i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (3.06 - 3.06i)T - 7iT^{2} \)
11 \( 1 + (1.51 + 3.66i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.88 + 0.780i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 4.54iT - 17T^{2} \)
19 \( 1 + (0.534 + 0.221i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.41 + 4.41i)T + 23iT^{2} \)
29 \( 1 + (1.96 - 4.74i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 0.0539T + 31T^{2} \)
37 \( 1 + (-0.798 + 0.330i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.621 + 0.621i)T + 41iT^{2} \)
43 \( 1 + (-0.857 - 2.06i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 9.44iT - 47T^{2} \)
53 \( 1 + (4.16 + 10.0i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (7.17 - 2.97i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (4.02 - 9.72i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-3.12 + 7.53i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-2.99 + 2.99i)T - 71iT^{2} \)
73 \( 1 + (-2.91 - 2.91i)T + 73iT^{2} \)
79 \( 1 - 5.74iT - 79T^{2} \)
83 \( 1 + (-3.30 - 1.36i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (2.38 - 2.38i)T - 89iT^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−9.127817637130109204679180055916, −8.449815792026215565010847102373, −7.911134927358492110661499297595, −6.85477054751205134840616341721, −6.16460475128297253916872472260, −5.54199925646601537547582428763, −3.57273697473508882973942461691, −2.80779409916944081650499468209, −1.91242838900410131360155017352, −0.12677025860278484061920272270, 2.45539010972279177873502913124, 3.47606929812385169696594632705, 4.25175323236528842302918412946, 4.84156165209855898246741483529, 6.15047692400701116827367611442, 7.37729827328524039342699926143, 7.87350643259417124946918189726, 9.315079795314309695951887097003, 9.666801160627235417249888015770, 10.10794903634694694662854343760

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