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Properties

Label 2-20-4.3-c8-0-5
Degree $2$
Conductor $20$
Sign $0.749 - 0.661i$
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  + (−14.9 + 5.65i)2-s + 25.1i·3-s + (191. − 169. i)4-s − 279.·5-s + (−142. − 376. i)6-s − 2.97e3i·7-s + (−1.91e3 + 3.62e3i)8-s + 5.92e3·9-s + (4.18e3 − 1.58e3i)10-s + 2.80e4i·11-s + (4.25e3 + 4.82e3i)12-s + 3.56e4·13-s + (1.68e4 + 4.45e4i)14-s − 7.02e3i·15-s + (8.16e3 − 6.50e4i)16-s + 5.93e4·17-s + ⋯
L(s)  = 1  + (−0.935 + 0.353i)2-s + 0.310i·3-s + (0.749 − 0.661i)4-s − 0.447·5-s + (−0.109 − 0.290i)6-s − 1.23i·7-s + (−0.467 + 0.884i)8-s + 0.903·9-s + (0.418 − 0.158i)10-s + 1.91i·11-s + (0.205 + 0.232i)12-s + 1.24·13-s + (0.438 + 1.15i)14-s − 0.138i·15-s + (0.124 − 0.992i)16-s + 0.710·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.749 - 0.661i$
Analytic conductor: \(8.14757\)
Root analytic conductor: \(2.85439\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :4),\ 0.749 - 0.661i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.04520 + 0.395186i\)
\(L(\frac12)\) \(\approx\) \(1.04520 + 0.395186i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (14.9 - 5.65i)T \)
5 \( 1 + 279.T \)
good3 \( 1 - 25.1iT - 6.56e3T^{2} \)
7 \( 1 + 2.97e3iT - 5.76e6T^{2} \)
11 \( 1 - 2.80e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.56e4T + 8.15e8T^{2} \)
17 \( 1 - 5.93e4T + 6.97e9T^{2} \)
19 \( 1 - 1.46e5iT - 1.69e10T^{2} \)
23 \( 1 + 3.09e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.02e6T + 5.00e11T^{2} \)
31 \( 1 + 1.62e5iT - 8.52e11T^{2} \)
37 \( 1 - 8.64e5T + 3.51e12T^{2} \)
41 \( 1 - 1.51e6T + 7.98e12T^{2} \)
43 \( 1 + 2.39e6iT - 1.16e13T^{2} \)
47 \( 1 - 2.15e6iT - 2.38e13T^{2} \)
53 \( 1 + 9.78e6T + 6.22e13T^{2} \)
59 \( 1 + 7.13e6iT - 1.46e14T^{2} \)
61 \( 1 + 5.43e6T + 1.91e14T^{2} \)
67 \( 1 - 1.02e7iT - 4.06e14T^{2} \)
71 \( 1 - 3.55e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.90e6T + 8.06e14T^{2} \)
79 \( 1 + 3.17e6iT - 1.51e15T^{2} \)
83 \( 1 - 4.52e7iT - 2.25e15T^{2} \)
89 \( 1 - 4.70e7T + 3.93e15T^{2} \)
97 \( 1 + 1.22e7T + 7.83e15T^{2} \)
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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

−16.56267254693024473907957961936, −15.64371448146301616590756349020, −14.39585944880952084004979587671, −12.44772181611498159403411805519, −10.60957404340514994707054296839, −9.859372154273360286427776887838, −7.925327196024062147325147752034, −6.81432798255040991549460421299, −4.28348006125665235449125993286, −1.24842739707794167449920408138, 1.01668248152558800710877177979, 3.16383705402529749390716506159, 6.20915077216700571498840446476, 8.056798358649719222998068384705, 9.090593474457492659445240267851, 10.95038756131101088911212692757, 11.96059487726770277840100378225, 13.37217579097446640650199369000, 15.60901203794583898149424166990, 16.18110862334797127210625166431

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