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Properties

Label 2-2e10-32.5-c1-0-13
Degree $2$
Conductor $1024$
Sign $0.896 + 0.442i$
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  + (−0.356 − 0.860i)3-s + (−0.883 − 0.366i)5-s + (2.35 + 2.35i)7-s + (1.50 − 1.50i)9-s + (0.752 − 1.81i)11-s + (−2.04 + 0.848i)13-s + 0.891i·15-s + 6.00i·17-s + (3.73 − 1.54i)19-s + (1.18 − 2.86i)21-s + (2.91 − 2.91i)23-s + (−2.88 − 2.88i)25-s + (−4.41 − 1.82i)27-s + (4.01 + 9.69i)29-s + 7.52·31-s + ⋯
L(s)  = 1  + (−0.205 − 0.497i)3-s + (−0.395 − 0.163i)5-s + (0.889 + 0.889i)7-s + (0.502 − 0.502i)9-s + (0.226 − 0.547i)11-s + (−0.568 + 0.235i)13-s + 0.230i·15-s + 1.45i·17-s + (0.856 − 0.354i)19-s + (0.258 − 0.624i)21-s + (0.607 − 0.607i)23-s + (−0.577 − 0.577i)25-s + (−0.850 − 0.352i)27-s + (0.745 + 1.80i)29-s + 1.35·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.896 + 0.442i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.896 + 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.629014716\)
\(L(\frac12)\) \(\approx\) \(1.629014716\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.356 + 0.860i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.883 + 0.366i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.35 - 2.35i)T + 7iT^{2} \)
11 \( 1 + (-0.752 + 1.81i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.04 - 0.848i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 6.00iT - 17T^{2} \)
19 \( 1 + (-3.73 + 1.54i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.91 + 2.91i)T - 23iT^{2} \)
29 \( 1 + (-4.01 - 9.69i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 7.52T + 31T^{2} \)
37 \( 1 + (2.40 + 0.994i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.37 + 1.37i)T - 41iT^{2} \)
43 \( 1 + (-4.50 + 10.8i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 3.33iT - 47T^{2} \)
53 \( 1 + (-3.32 + 8.02i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-11.3 - 4.70i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-0.0688 - 0.166i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (3.47 + 8.39i)T + (-47.3 + 47.3i)T^{2} \)
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 \( 1 + (4.79 - 1.98i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (6.38 + 6.38i)T + 89iT^{2} \)
97 \( 1 - 0.874T + 97T^{2} \)
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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

−9.893978861454843750428538287316, −8.726183977884942546124704257258, −8.423618092051327271656274778253, −7.32907752053230386621321248413, −6.55466368852679642682445043810, −5.59939471164106240778608000794, −4.70423157175385443977548999240, −3.63677581547551101818106223936, −2.24864338762012116625281486527, −1.03598246454569126653804840236, 1.11471179319599125210495191211, 2.66367922932548682481257777899, 4.05479975107747202819635535424, 4.66623368912938830998579914602, 5.40308313954852650557128579594, 6.89936762901917459579553298784, 7.58853629291710304357864355077, 8.018033407619609320932174907316, 9.654525983320310215205755012964, 9.803071237491664127267567031398

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