↑

Properties

Label 2-2e10-32.29-c1-0-26
Degree $2$
Conductor $1024$
Sign $-0.659 + 0.751i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.860 − 0.356i)3-s + (−0.366 − 0.883i)5-s + (2.35 − 2.35i)7-s + (−1.50 − 1.50i)9-s + (1.81 − 0.752i)11-s + (−0.848 + 2.04i)13-s + 0.891i·15-s − 6.00i·17-s + (−1.54 + 3.73i)19-s + (−2.86 + 1.18i)21-s + (2.91 + 2.91i)23-s + (2.88 − 2.88i)25-s + (1.82 + 4.41i)27-s + (−9.69 − 4.01i)29-s − 7.52·31-s + ⋯
L(s)  = 1  + (−0.497 − 0.205i)3-s + (−0.163 − 0.395i)5-s + (0.889 − 0.889i)7-s + (−0.502 − 0.502i)9-s + (0.547 − 0.226i)11-s + (−0.235 + 0.568i)13-s + 0.230i·15-s − 1.45i·17-s + (−0.354 + 0.856i)19-s + (−0.624 + 0.258i)21-s + (0.607 + 0.607i)23-s + (0.577 − 0.577i)25-s + (0.352 + 0.850i)27-s + (−1.80 − 0.745i)29-s − 1.35·31-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.024784283\)
\(L(\frac12)\) \(\approx\) \(1.024784283\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.860 + 0.356i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.366 + 0.883i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.35 + 2.35i)T - 7iT^{2} \)
11 \( 1 + (-1.81 + 0.752i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (0.848 - 2.04i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 6.00iT - 17T^{2} \)
19 \( 1 + (1.54 - 3.73i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.91 - 2.91i)T + 23iT^{2} \)
29 \( 1 + (9.69 + 4.01i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 7.52T + 31T^{2} \)
37 \( 1 + (0.994 + 2.40i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.37 + 1.37i)T + 41iT^{2} \)
43 \( 1 + (-10.8 + 4.50i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 3.33iT - 47T^{2} \)
53 \( 1 + (8.02 - 3.32i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (4.70 + 11.3i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (0.166 + 0.0688i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (8.39 + 3.47i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-7.92 + 7.92i)T - 71iT^{2} \)
73 \( 1 + (5.84 + 5.84i)T + 73iT^{2} \)
79 \( 1 + 1.80iT - 79T^{2} \)
83 \( 1 + (-1.98 + 4.79i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-6.38 + 6.38i)T - 89iT^{2} \)
97 \( 1 - 0.874T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.351712269324484730473534226359, −9.062956529860876129539473988787, −7.74199780461699557483294004332, −7.26617525963055363266192993019, −6.20452464741766274623489964747, −5.28736342736548576668629920087, −4.40333947855528741008815673353, −3.46200286884123333852213846787, −1.75728512786060875524401458811, −0.49881228538480653106413510910, 1.74329708134306540412101539375, 2.88147993450135134533158643994, 4.19513551417704346901068712623, 5.23086191833210990353270336753, 5.75280885564088170092281540019, 6.85867253924293807724717123995, 7.80085198068374088387092560105, 8.652798336062506862368671442100, 9.260840289157033024882024176942, 10.65817331684835556164444050019

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