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Properties

Label 2-20-20.19-c10-0-21
Degree $2$
Conductor $20$
Sign $-0.0125 + 0.999i$
Analytic cond. $12.7071$
Root an. cond. $3.56470$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−23.9 + 21.2i)2-s + 94.4·3-s + (120. − 1.01e3i)4-s + (3.10e3 − 328. i)5-s + (−2.25e3 + 2.00e3i)6-s − 2.00e4·7-s + (1.87e4 + 2.68e4i)8-s − 5.01e4·9-s + (−6.73e4 + 7.39e4i)10-s − 1.98e5i·11-s + (1.13e4 − 9.60e4i)12-s + 3.52e5i·13-s + (4.80e5 − 4.26e5i)14-s + (2.93e5 − 3.10e4i)15-s + (−1.01e6 − 2.45e5i)16-s − 1.97e6i·17-s + ⋯
L(s)  = 1  + (−0.747 + 0.664i)2-s + 0.388·3-s + (0.117 − 0.993i)4-s + (0.994 − 0.105i)5-s + (−0.290 + 0.258i)6-s − 1.19·7-s + (0.571 + 0.820i)8-s − 0.848·9-s + (−0.673 + 0.739i)10-s − 1.23i·11-s + (0.0457 − 0.385i)12-s + 0.950i·13-s + (0.893 − 0.793i)14-s + (0.386 − 0.0408i)15-s + (−0.972 − 0.233i)16-s − 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0125 + 0.999i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.0125 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.0125 + 0.999i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ -0.0125 + 0.999i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.534380 - 0.541138i\)
\(L(\frac12)\) \(\approx\) \(0.534380 - 0.541138i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (23.9 - 21.2i)T \)
5 \( 1 + (-3.10e3 + 328. i)T \)
good3 \( 1 - 94.4T + 5.90e4T^{2} \)
7 \( 1 + 2.00e4T + 2.82e8T^{2} \)
11 \( 1 + 1.98e5iT - 2.59e10T^{2} \)
13 \( 1 - 3.52e5iT - 1.37e11T^{2} \)
17 \( 1 + 1.97e6iT - 2.01e12T^{2} \)
19 \( 1 + 3.78e6iT - 6.13e12T^{2} \)
23 \( 1 + 4.41e6T + 4.14e13T^{2} \)
29 \( 1 + 1.04e7T + 4.20e14T^{2} \)
31 \( 1 - 1.45e7iT - 8.19e14T^{2} \)
37 \( 1 + 1.27e8iT - 4.80e15T^{2} \)
41 \( 1 - 9.37e7T + 1.34e16T^{2} \)
43 \( 1 + 2.01e8T + 2.16e16T^{2} \)
47 \( 1 - 6.61e7T + 5.25e16T^{2} \)
53 \( 1 - 8.60e7iT - 1.74e17T^{2} \)
59 \( 1 - 5.09e8iT - 5.11e17T^{2} \)
61 \( 1 + 2.18e8T + 7.13e17T^{2} \)
67 \( 1 - 7.20e8T + 1.82e18T^{2} \)
71 \( 1 - 1.16e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.07e9iT - 4.29e18T^{2} \)
79 \( 1 - 3.63e9iT - 9.46e18T^{2} \)
83 \( 1 - 1.43e9T + 1.55e19T^{2} \)
89 \( 1 + 9.94e8T + 3.11e19T^{2} \)
97 \( 1 + 4.59e9iT - 7.37e19T^{2} \)
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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

−15.99596884798652755277101519727, −14.19261040219713360003116710279, −13.53803575468252459773775094898, −11.18955878015197430757284565504, −9.494427762461827897741793771592, −8.882137193272915326701691119814, −6.81447857767203008225454637393, −5.62142310467334440050012955460, −2.61941981216890222119200070384, −0.37086156428109815791256793783, 1.91485803755703067696853619146, 3.35337771045971351286534417425, 6.19302431074759259337637648492, 8.111500726671861071878844777388, 9.646467678185486239324849675043, 10.31878377930747251114649083855, 12.36358753692307878412037991851, 13.28219407079495258704524343362, 14.95273554080932317281208506954, 16.70899720326586465630273117399

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