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Properties

Label 2-20-5.4-c11-0-3
Degree $2$
Conductor $20$
Sign $0.924 - 0.380i$
Analytic cond. $15.3668$
Root an. cond. $3.92005$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 444. i·3-s + (6.46e3 − 2.65e3i)5-s − 5.64e4i·7-s − 2.01e4·9-s + 3.83e5·11-s + 1.21e6i·13-s + (1.18e6 + 2.87e6i)15-s − 4.49e6i·17-s + 1.68e7·19-s + 2.50e7·21-s + 4.22e7i·23-s + (3.46e7 − 3.43e7i)25-s + 6.97e7i·27-s + 2.58e7·29-s + 2.62e7·31-s + ⋯
L(s)  = 1  + 1.05i·3-s + (0.924 − 0.380i)5-s − 1.27i·7-s − 0.113·9-s + 0.717·11-s + 0.907i·13-s + (0.401 + 0.975i)15-s − 0.768i·17-s + 1.56·19-s + 1.34·21-s + 1.36i·23-s + (0.710 − 0.703i)25-s + 0.935i·27-s + 0.234·29-s + 0.164·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.924 - 0.380i$
Analytic conductor: \(15.3668\)
Root analytic conductor: \(3.92005\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :11/2),\ 0.924 - 0.380i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.29657 + 0.454076i\)
\(L(\frac12)\) \(\approx\) \(2.29657 + 0.454076i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-6.46e3 + 2.65e3i)T \)
good3 \( 1 - 444. iT - 1.77e5T^{2} \)
7 \( 1 + 5.64e4iT - 1.97e9T^{2} \)
11 \( 1 - 3.83e5T + 2.85e11T^{2} \)
13 \( 1 - 1.21e6iT - 1.79e12T^{2} \)
17 \( 1 + 4.49e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.68e7T + 1.16e14T^{2} \)
23 \( 1 - 4.22e7iT - 9.52e14T^{2} \)
29 \( 1 - 2.58e7T + 1.22e16T^{2} \)
31 \( 1 - 2.62e7T + 2.54e16T^{2} \)
37 \( 1 + 5.75e8iT - 1.77e17T^{2} \)
41 \( 1 + 9.40e8T + 5.50e17T^{2} \)
43 \( 1 - 1.37e9iT - 9.29e17T^{2} \)
47 \( 1 + 1.31e9iT - 2.47e18T^{2} \)
53 \( 1 + 2.71e9iT - 9.26e18T^{2} \)
59 \( 1 - 6.01e9T + 3.01e19T^{2} \)
61 \( 1 - 4.12e9T + 4.35e19T^{2} \)
67 \( 1 - 1.12e9iT - 1.22e20T^{2} \)
71 \( 1 - 8.09e9T + 2.31e20T^{2} \)
73 \( 1 + 3.17e10iT - 3.13e20T^{2} \)
79 \( 1 + 3.33e10T + 7.47e20T^{2} \)
83 \( 1 - 3.13e10iT - 1.28e21T^{2} \)
89 \( 1 + 1.00e11T + 2.77e21T^{2} \)
97 \( 1 - 4.33e10iT - 7.15e21T^{2} \)
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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

−16.05867313224212531046604547514, −14.30102332469444823817548472681, −13.48586434491836484546615847260, −11.48812091105378416567900592065, −9.983272650284278402767606537277, −9.331377326812906807787488786869, −7.05446974988637239204326690986, −5.09058016862383911430121911101, −3.76029389142707580816009947816, −1.25518440401479639809979801157, 1.32497785102783206044997375307, 2.68979670257792901764327326162, 5.61740411262303805193221940871, 6.76170381441183482335656225539, 8.490189037011683612440565783956, 10.03466815877346074323151768865, 11.91334294411615794114146470299, 12.90202579109226033477278193731, 14.13435438866553391764533720996, 15.41962904178212584166718492923

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