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Properties

Label 2-800-40.19-c4-0-59
Degree $2$
Conductor $800$
Sign $-0.714 + 0.700i$
Analytic cond. $82.6959$
Root an. cond. $9.09373$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.2i·3-s + 43.3·7-s − 23.1·9-s + 165.·11-s − 201.·13-s + 172. i·17-s − 640.·19-s − 441. i·21-s + 497.·23-s − 590. i·27-s − 509. i·29-s − 492. i·31-s − 1.68e3i·33-s + 678.·37-s + 2.05e3i·39-s + ⋯
L(s)  = 1  − 1.13i·3-s + 0.883·7-s − 0.285·9-s + 1.36·11-s − 1.19·13-s + 0.597i·17-s − 1.77·19-s − 1.00i·21-s + 0.940·23-s − 0.810i·27-s − 0.606i·29-s − 0.512i·31-s − 1.54i·33-s + 0.495·37-s + 1.35i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.700i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.714 + 0.700i$
Analytic conductor: \(82.6959\)
Root analytic conductor: \(9.09373\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :2),\ -0.714 + 0.700i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.047364764\)
\(L(\frac12)\) \(\approx\) \(2.047364764\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 10.2iT - 81T^{2} \)
7 \( 1 - 43.3T + 2.40e3T^{2} \)
11 \( 1 - 165.T + 1.46e4T^{2} \)
13 \( 1 + 201.T + 2.85e4T^{2} \)
17 \( 1 - 172. iT - 8.35e4T^{2} \)
19 \( 1 + 640.T + 1.30e5T^{2} \)
23 \( 1 - 497.T + 2.79e5T^{2} \)
29 \( 1 + 509. iT - 7.07e5T^{2} \)
31 \( 1 + 492. iT - 9.23e5T^{2} \)
37 \( 1 - 678.T + 1.87e6T^{2} \)
41 \( 1 + 613.T + 2.82e6T^{2} \)
43 \( 1 + 1.39e3iT - 3.41e6T^{2} \)
47 \( 1 - 965.T + 4.87e6T^{2} \)
53 \( 1 - 4.61e3T + 7.89e6T^{2} \)
59 \( 1 + 822.T + 1.21e7T^{2} \)
61 \( 1 + 6.91e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.92e3iT - 2.01e7T^{2} \)
71 \( 1 + 4.23e3iT - 2.54e7T^{2} \)
73 \( 1 - 8.50e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.02e4iT - 3.89e7T^{2} \)
83 \( 1 + 818. iT - 4.74e7T^{2} \)
89 \( 1 - 1.04e4T + 6.27e7T^{2} \)
97 \( 1 + 1.16e4iT - 8.85e7T^{2} \)
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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

−9.199615041360660596559620309586, −8.373735070085343036786289311816, −7.60241237695705864767860519251, −6.79095253603835812549211774312, −6.15156838463631012641659535679, −4.81832803346837291454019053832, −3.97779549028878599894719347715, −2.29717707242772159652673357047, −1.64444268524952570416178598654, −0.47854252105765561282699434614, 1.23432627196308448692049578344, 2.54962649969273788105077300777, 3.91024796753645282885444829957, 4.56852342436524208323713858479, 5.24626943964395500232285996501, 6.59814886274430914319353241886, 7.38412946074297020320509469152, 8.659032237291636822832427827387, 9.135216827976911894156515851170, 10.03518034208551245442515635389

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