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Properties

Label 2-20-5.3-c8-0-3
Degree $2$
Conductor $20$
Sign $-0.487 - 0.873i$
Analytic cond. $8.14757$
Root an. cond. $2.85439$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−103. − 103. i)3-s + (304. − 545. i)5-s + (−1.62e3 + 1.62e3i)7-s + 1.49e4i·9-s − 7.35e3·11-s + (4.12e3 + 4.12e3i)13-s + (−8.82e4 + 2.50e4i)15-s + (3.63e4 − 3.63e4i)17-s + 2.46e5i·19-s + 3.37e5·21-s + (−3.51e5 − 3.51e5i)23-s + (−2.05e5 − 3.32e5i)25-s + (8.74e5 − 8.74e5i)27-s + 1.24e5i·29-s − 5.20e5·31-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)3-s + (0.486 − 0.873i)5-s + (−0.677 + 0.677i)7-s + 2.28i·9-s − 0.502·11-s + (0.144 + 0.144i)13-s + (−1.74 + 0.495i)15-s + (0.434 − 0.434i)17-s + 1.89i·19-s + 1.73·21-s + (−1.25 − 1.25i)23-s + (−0.526 − 0.850i)25-s + (1.64 − 1.64i)27-s + 0.175i·29-s − 0.563·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.487 - 0.873i$
Analytic conductor: \(8.14757\)
Root analytic conductor: \(2.85439\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :4),\ -0.487 - 0.873i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0300131 + 0.0511131i\)
\(L(\frac12)\) \(\approx\) \(0.0300131 + 0.0511131i\)
\(L(5)\) 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 \( 1 + (-304. + 545. i)T \)
good3 \( 1 + (103. + 103. i)T + 6.56e3iT^{2} \)
7 \( 1 + (1.62e3 - 1.62e3i)T - 5.76e6iT^{2} \)
11 \( 1 + 7.35e3T + 2.14e8T^{2} \)
13 \( 1 + (-4.12e3 - 4.12e3i)T + 8.15e8iT^{2} \)
17 \( 1 + (-3.63e4 + 3.63e4i)T - 6.97e9iT^{2} \)
19 \( 1 - 2.46e5iT - 1.69e10T^{2} \)
23 \( 1 + (3.51e5 + 3.51e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 1.24e5iT - 5.00e11T^{2} \)
31 \( 1 + 5.20e5T + 8.52e11T^{2} \)
37 \( 1 + (1.16e6 - 1.16e6i)T - 3.51e12iT^{2} \)
41 \( 1 + 3.72e6T + 7.98e12T^{2} \)
43 \( 1 + (2.86e6 + 2.86e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (2.45e6 - 2.45e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-4.77e6 - 4.77e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 7.42e6iT - 1.46e14T^{2} \)
61 \( 1 + 5.46e6T + 1.91e14T^{2} \)
67 \( 1 + (-1.14e7 + 1.14e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 2.65e7T + 6.45e14T^{2} \)
73 \( 1 + (1.87e7 + 1.87e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 1.79e6iT - 1.51e15T^{2} \)
83 \( 1 + (-1.15e7 - 1.15e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 3.57e7iT - 3.93e15T^{2} \)
97 \( 1 + (3.39e7 - 3.39e7i)T - 7.83e15iT^{2} \)
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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

−16.19894347308741904874589513630, −13.78713565578827675515268020078, −12.52705285126997001471990963167, −12.09865022887318801847740047718, −10.18132606374887707215866094765, −8.163918178785091676041877191690, −6.33977016625311350513588413138, −5.38230094842907061035518594469, −1.75865394334544675892727432177, −0.03432330432509585533609658412, 3.59520153066142442388506976854, 5.42206231285029482342443156800, 6.77200290341409534084951023465, 9.692736908895466973907684544700, 10.46407458652403742204903795580, 11.50423424537105404417342769076, 13.40149664974893834594239622307, 15.12890975774447637607785854369, 16.06473928225373534621156394864, 17.23712851934775937974057704895

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