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Properties

Label 2-20-20.3-c9-0-22
Degree $2$
Conductor $20$
Sign $-0.316 + 0.948i$
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  + (15.5 − 16.4i)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 + (−8.83e3 − 7.49e3i)8-s − 1.93e4i·9-s + (1.73e4 − 2.64e4i)10-s + 9.40e4i·11-s + (−7.53e4 − 6.75e4i)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  + (0.687 − 0.726i)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 + (−0.762 − 0.647i)8-s − 0.984i·9-s + (0.548 − 0.836i)10-s + 1.93i·11-s + (−1.04 − 0.940i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.316 + 0.948i$
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: \((2,\ 20,\ (\ :9/2),\ -0.316 + 0.948i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.17883 - 3.02338i\)
\(L(\frac12)\) \(\approx\) \(2.17883 - 3.02338i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-15.5 + 16.4i)T \)
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

−14.95017553270460539789419471031, −14.37504184817870296356732036835, −12.90178158723007307516798146454, −12.47734298700230848077488912703, −10.27355025979655716001838250938, −8.895696757762445065770014055507, −6.93826250972581452152120861458, −4.98793426205251059601509866188, −2.46168428767933609910716530961, −1.69127195627932417388214455995, 2.83529345698877489697860091567, 4.35107342429100360659103723142, 6.09003106032281655795562113781, 8.172117023046972205264832943557, 9.362209325247599948068847959751, 11.06239485408839792767083640518, 13.39431110196826988820518806133, 14.13937185218039270128234034166, 14.98102623774645855981251582551, 16.40360061193333632826029708829

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