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Properties

Label 2-2e10-8.5-c3-0-35
Degree $2$
Conductor $1024$
Sign $-i$
Analytic cond. $60.4179$
Root an. cond. $7.77289$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.80i·3-s − 0.844i·5-s + 29.0·7-s + 19.1·9-s − 17.1i·11-s + 68.6i·13-s + 2.36·15-s − 86.7·17-s + 77.5i·19-s + 81.5i·21-s − 70.2·23-s + 124.·25-s + 129. i·27-s + 89.6i·29-s − 8.86·31-s + ⋯
L(s)  = 1  + 0.539i·3-s − 0.0754i·5-s + 1.57·7-s + 0.708·9-s − 0.470i·11-s + 1.46i·13-s + 0.0407·15-s − 1.23·17-s + 0.936i·19-s + 0.847i·21-s − 0.636·23-s + 0.994·25-s + 0.922i·27-s + 0.574i·29-s − 0.0513·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-i$
Analytic conductor: \(60.4179\)
Root analytic conductor: \(7.77289\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (513, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :3/2),\ -i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.497685556\)
\(L(\frac12)\) \(\approx\) \(2.497685556\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.80iT - 27T^{2} \)
5 \( 1 + 0.844iT - 125T^{2} \)
7 \( 1 - 29.0T + 343T^{2} \)
11 \( 1 + 17.1iT - 1.33e3T^{2} \)
13 \( 1 - 68.6iT - 2.19e3T^{2} \)
17 \( 1 + 86.7T + 4.91e3T^{2} \)
19 \( 1 - 77.5iT - 6.85e3T^{2} \)
23 \( 1 + 70.2T + 1.21e4T^{2} \)
29 \( 1 - 89.6iT - 2.43e4T^{2} \)
31 \( 1 + 8.86T + 2.97e4T^{2} \)
37 \( 1 + 30.7iT - 5.06e4T^{2} \)
41 \( 1 - 153.T + 6.89e4T^{2} \)
43 \( 1 - 171. iT - 7.95e4T^{2} \)
47 \( 1 - 99.9T + 1.03e5T^{2} \)
53 \( 1 + 550. iT - 1.48e5T^{2} \)
59 \( 1 + 459. iT - 2.05e5T^{2} \)
61 \( 1 + 0.479iT - 2.26e5T^{2} \)
67 \( 1 - 799. iT - 3.00e5T^{2} \)
71 \( 1 + 419.T + 3.57e5T^{2} \)
73 \( 1 + 374.T + 3.89e5T^{2} \)
79 \( 1 - 705.T + 4.93e5T^{2} \)
83 \( 1 - 1.33e3iT - 5.71e5T^{2} \)
89 \( 1 + 4.72T + 7.04e5T^{2} \)
97 \( 1 - 379.T + 9.12e5T^{2} \)
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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

−9.737445270422097193420174811107, −8.887039967709211458271222085528, −8.282352069812546464574733717728, −7.28067221154129848601604551535, −6.43626384738752224433898080900, −5.18473303226904512577925546189, −4.48684730758625065186756909745, −3.85657448242218918873734626192, −2.14773733466127443854419669740, −1.32132562509647990123949139896, 0.64406394985347528742372269146, 1.74174066951988463662395439528, 2.66564811048867526520263158440, 4.30711647085303208476366433832, 4.85193768674634564743700512631, 5.94774148528880125541125746047, 7.05998859740731948985293498297, 7.64542659274569087060645306144, 8.359076232979020951801583323581, 9.229840789783708102582541576402

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