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Properties

Label 2-20-5.4-c9-0-2
Degree $2$
Conductor $20$
Sign $0.569 + 0.821i$
Analytic cond. $10.3007$
Root an. cond. $3.20947$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 92.1i·3-s + (−796. − 1.14e3i)5-s − 2.20e3i·7-s + 1.11e4·9-s + 4.12e4·11-s − 1.67e5i·13-s + (1.05e5 − 7.33e4i)15-s − 3.12e5i·17-s + 4.39e5·19-s + 2.03e5·21-s − 1.29e6i·23-s + (−6.84e5 + 1.82e6i)25-s + 2.84e6i·27-s − 2.74e6·29-s − 3.48e6·31-s + ⋯
L(s)  = 1  + 0.656i·3-s + (−0.569 − 0.821i)5-s − 0.347i·7-s + 0.568·9-s + 0.850·11-s − 1.62i·13-s + (0.539 − 0.374i)15-s − 0.906i·17-s + 0.773·19-s + 0.227·21-s − 0.963i·23-s + (−0.350 + 0.936i)25-s + 1.03i·27-s − 0.721·29-s − 0.677·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :9/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.35274 - 0.708167i\)
\(L(\frac12)\) \(\approx\) \(1.35274 - 0.708167i\)
\(L(\frac{11}{2})\) 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 \( 1 + (796. + 1.14e3i)T \)
good3 \( 1 - 92.1iT - 1.96e4T^{2} \)
7 \( 1 + 2.20e3iT - 4.03e7T^{2} \)
11 \( 1 - 4.12e4T + 2.35e9T^{2} \)
13 \( 1 + 1.67e5iT - 1.06e10T^{2} \)
17 \( 1 + 3.12e5iT - 1.18e11T^{2} \)
19 \( 1 - 4.39e5T + 3.22e11T^{2} \)
23 \( 1 + 1.29e6iT - 1.80e12T^{2} \)
29 \( 1 + 2.74e6T + 1.45e13T^{2} \)
31 \( 1 + 3.48e6T + 2.64e13T^{2} \)
37 \( 1 - 5.59e6iT - 1.29e14T^{2} \)
41 \( 1 + 2.77e6T + 3.27e14T^{2} \)
43 \( 1 + 1.61e7iT - 5.02e14T^{2} \)
47 \( 1 + 5.28e7iT - 1.11e15T^{2} \)
53 \( 1 - 5.94e7iT - 3.29e15T^{2} \)
59 \( 1 - 1.32e8T + 8.66e15T^{2} \)
61 \( 1 + 1.89e8T + 1.16e16T^{2} \)
67 \( 1 - 1.32e8iT - 2.72e16T^{2} \)
71 \( 1 + 4.51e7T + 4.58e16T^{2} \)
73 \( 1 - 1.44e8iT - 5.88e16T^{2} \)
79 \( 1 - 5.87e8T + 1.19e17T^{2} \)
83 \( 1 + 7.24e8iT - 1.86e17T^{2} \)
89 \( 1 - 1.07e9T + 3.50e17T^{2} \)
97 \( 1 + 1.05e8iT - 7.60e17T^{2} \)
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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

−16.00134647225276260286361439566, −14.98357456852874547243165013322, −13.25632670201186518593292826925, −11.98351115407716587155092486028, −10.40696779093953260559744388326, −9.063392337495464911347893186160, −7.45762597644838548312978382175, −5.12610363506469404371186895216, −3.71092890107768823603798656780, −0.78172087094473610666326816353, 1.71863852892113351256218250222, 3.90062114571439388228656887930, 6.44260802503671268791772635036, 7.53572376205453944662711563450, 9.393034218773736053742133713174, 11.24849808607303807519580168606, 12.25558412649317231770617232614, 13.84271117612176206262864703587, 14.98767161687828995590146978714, 16.34045022336025943219213743405

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