Properties

Label 2-20-20.3-c9-0-18
Degree $2$
Conductor $20$
Sign $-0.417 + 0.908i$
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  + (16.4 − 15.5i)2-s + (−139. + 139. i)3-s + (27.7 − 511. i)4-s + (1.37e3 − 246. i)5-s + (−121. + 4.47e3i)6-s + (−3.05e3 − 3.05e3i)7-s + (−7.49e3 − 8.83e3i)8-s − 1.93e4i·9-s + (1.87e4 − 2.54e4i)10-s − 9.40e4i·11-s + (6.75e4 + 7.53e4i)12-s + (−5.71e4 − 5.71e4i)13-s + (−9.76e4 − 2.64e3i)14-s + (−1.57e5 + 2.26e5i)15-s + (−2.60e5 − 2.83e4i)16-s + (1.91e4 − 1.91e4i)17-s + ⋯
L(s)  = 1  + (0.726 − 0.687i)2-s + (−0.996 + 0.996i)3-s + (0.0541 − 0.998i)4-s + (0.984 − 0.176i)5-s + (−0.0381 + 1.40i)6-s + (−0.480 − 0.480i)7-s + (−0.647 − 0.762i)8-s − 0.984i·9-s + (0.593 − 0.805i)10-s − 1.93i·11-s + (0.940 + 1.04i)12-s + (−0.554 − 0.554i)13-s + (−0.679 − 0.0184i)14-s + (−0.804 + 1.15i)15-s + (−0.994 − 0.108i)16-s + (0.0555 − 0.0555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.417 + 0.908i$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :9/2),\ -0.417 + 0.908i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.877780 - 1.36865i\)
\(L(\frac12)\) \(\approx\) \(0.877780 - 1.36865i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-16.4 + 15.5i)T \)
5 \( 1 + (-1.37e3 + 246. i)T \)
good3 \( 1 + (139. - 139. i)T - 1.96e4iT^{2} \)
7 \( 1 + (3.05e3 + 3.05e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 9.40e4iT - 2.35e9T^{2} \)
13 \( 1 + (5.71e4 + 5.71e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-1.91e4 + 1.91e4i)T - 1.18e11iT^{2} \)
19 \( 1 - 6.74e5T + 3.22e11T^{2} \)
23 \( 1 + (5.76e5 - 5.76e5i)T - 1.80e12iT^{2} \)
29 \( 1 - 4.76e6iT - 1.45e13T^{2} \)
31 \( 1 + 4.09e6iT - 2.64e13T^{2} \)
37 \( 1 + (-6.25e6 + 6.25e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 5.40e6T + 3.27e14T^{2} \)
43 \( 1 + (2.13e7 - 2.13e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (-2.28e7 - 2.28e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (2.40e7 + 2.40e7i)T + 3.29e15iT^{2} \)
59 \( 1 + 7.68e6T + 8.66e15T^{2} \)
61 \( 1 - 8.22e6T + 1.16e16T^{2} \)
67 \( 1 + (-1.24e8 - 1.24e8i)T + 2.72e16iT^{2} \)
71 \( 1 + 2.02e8iT - 4.58e16T^{2} \)
73 \( 1 + (-9.32e7 - 9.32e7i)T + 5.88e16iT^{2} \)
79 \( 1 - 3.87e8T + 1.19e17T^{2} \)
83 \( 1 + (-1.97e7 + 1.97e7i)T - 1.86e17iT^{2} \)
89 \( 1 + 9.93e8iT - 3.50e17T^{2} \)
97 \( 1 + (4.40e8 - 4.40e8i)T - 7.60e17iT^{2} \)
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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

−15.95688731844060574930338444616, −14.19484418907079016712904489139, −13.14513333597069927918344167874, −11.49026117550504565688375830097, −10.47701576730124253042253434170, −9.525223664505286276656867411082, −6.04421212844368265834473315119, −5.18626779162806953274104210391, −3.32778712512459120292337473491, −0.67047453478045913122018324205, 2.13749337393413948324111007001, 5.07619070577228551830583388048, 6.36581296433785366845103792423, 7.26276591116368349428899151782, 9.676102370313344780661046959192, 11.95212808852618847310126908851, 12.59997934506911303471180285415, 13.82080484064716649725684975170, 15.23509511038066768311653207079, 16.83569765936554744064085040015

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