Properties

Label 2-2e10-8.5-c3-0-88
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  − 4.62i·3-s − 17.8i·5-s − 13.8·7-s + 5.59·9-s − 2.18i·11-s − 46.3i·13-s − 82.7·15-s − 18.6·17-s + 122. i·19-s + 64.1i·21-s − 134.·23-s − 194.·25-s − 150. i·27-s − 84.5i·29-s − 31.5·31-s + ⋯
L(s)  = 1  − 0.890i·3-s − 1.59i·5-s − 0.749·7-s + 0.207·9-s − 0.0599i·11-s − 0.989i·13-s − 1.42·15-s − 0.266·17-s + 1.47i·19-s + 0.667i·21-s − 1.21·23-s − 1.55·25-s − 1.07i·27-s − 0.541i·29-s − 0.182·31-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

\(L(2)\) \(\approx\) \(0.4159824014\)
\(L(\frac12)\) \(\approx\) \(0.4159824014\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.62iT - 27T^{2} \)
5 \( 1 + 17.8iT - 125T^{2} \)
7 \( 1 + 13.8T + 343T^{2} \)
11 \( 1 + 2.18iT - 1.33e3T^{2} \)
13 \( 1 + 46.3iT - 2.19e3T^{2} \)
17 \( 1 + 18.6T + 4.91e3T^{2} \)
19 \( 1 - 122. iT - 6.85e3T^{2} \)
23 \( 1 + 134.T + 1.21e4T^{2} \)
29 \( 1 + 84.5iT - 2.43e4T^{2} \)
31 \( 1 + 31.5T + 2.97e4T^{2} \)
37 \( 1 - 126. iT - 5.06e4T^{2} \)
41 \( 1 + 210.T + 6.89e4T^{2} \)
43 \( 1 + 168. iT - 7.95e4T^{2} \)
47 \( 1 - 182.T + 1.03e5T^{2} \)
53 \( 1 - 37.0iT - 1.48e5T^{2} \)
59 \( 1 - 624. iT - 2.05e5T^{2} \)
61 \( 1 + 246. iT - 2.26e5T^{2} \)
67 \( 1 - 129. iT - 3.00e5T^{2} \)
71 \( 1 + 348.T + 3.57e5T^{2} \)
73 \( 1 + 299.T + 3.89e5T^{2} \)
79 \( 1 - 943.T + 4.93e5T^{2} \)
83 \( 1 + 443. iT - 5.71e5T^{2} \)
89 \( 1 + 1.41e3T + 7.04e5T^{2} \)
97 \( 1 - 1.51e3T + 9.12e5T^{2} \)
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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

−8.747807040402326329020025411935, −8.106137529684470556688395679255, −7.45154164376702118690564519038, −6.22566005777597742026995183148, −5.67093780219743106740325979147, −4.53759942600252357630155214016, −3.57292140568406433679999687611, −2.03714013342576036549107376016, −1.09109862497464628654397917523, −0.10925737787235762421973151645, 2.08697124004289894218962419174, 3.12797331220385186908131125349, 3.88043137484788058246384999060, 4.81026649157314861280491380484, 6.15232530021591832749580873146, 6.81712803476495746275377785807, 7.38347780405096529477203116757, 8.795347286511747317533939541615, 9.633468796054178963694805071780, 10.07390204957860609751471696254

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