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Properties

Label 2-20-20.3-c9-0-13
Degree $2$
Conductor $20$
Sign $0.128 - 0.991i$
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  + (21.6 + 6.66i)2-s + (−40.3 + 40.3i)3-s + (423. + 288. i)4-s + (1.19e3 + 728. i)5-s + (−1.14e3 + 603. i)6-s + (−3.88e3 − 3.88e3i)7-s + (7.23e3 + 9.04e3i)8-s + 1.64e4i·9-s + (2.09e4 + 2.36e4i)10-s + 4.79e4i·11-s + (−2.86e4 + 5.45e3i)12-s + (3.41e4 + 3.41e4i)13-s + (−5.81e4 − 1.09e5i)14-s + (−7.74e4 + 1.87e4i)15-s + (9.61e4 + 2.43e5i)16-s + (2.44e5 − 2.44e5i)17-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)2-s + (−0.287 + 0.287i)3-s + (0.826 + 0.562i)4-s + (0.853 + 0.521i)5-s + (−0.359 + 0.190i)6-s + (−0.612 − 0.612i)7-s + (0.624 + 0.781i)8-s + 0.834i·9-s + (0.662 + 0.749i)10-s + 0.986i·11-s + (−0.399 + 0.0759i)12-s + (0.331 + 0.331i)13-s + (−0.404 − 0.765i)14-s + (−0.395 + 0.0955i)15-s + (0.366 + 0.930i)16-s + (0.710 − 0.710i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.128 - 0.991i$
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: \((2,\ 20,\ (\ :9/2),\ 0.128 - 0.991i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.27040 + 1.99572i\)
\(L(\frac12)\) \(\approx\) \(2.27040 + 1.99572i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-21.6 - 6.66i)T \)
5 \( 1 + (-1.19e3 - 728. i)T \)
good3 \( 1 + (40.3 - 40.3i)T - 1.96e4iT^{2} \)
7 \( 1 + (3.88e3 + 3.88e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 4.79e4iT - 2.35e9T^{2} \)
13 \( 1 + (-3.41e4 - 3.41e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-2.44e5 + 2.44e5i)T - 1.18e11iT^{2} \)
19 \( 1 + 5.25e5T + 3.22e11T^{2} \)
23 \( 1 + (-1.02e6 + 1.02e6i)T - 1.80e12iT^{2} \)
29 \( 1 + 4.16e6iT - 1.45e13T^{2} \)
31 \( 1 + 9.41e6iT - 2.64e13T^{2} \)
37 \( 1 + (-6.06e5 + 6.06e5i)T - 1.29e14iT^{2} \)
41 \( 1 - 2.53e7T + 3.27e14T^{2} \)
43 \( 1 + (5.84e6 - 5.84e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (-1.42e7 - 1.42e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (-1.67e6 - 1.67e6i)T + 3.29e15iT^{2} \)
59 \( 1 + 2.56e6T + 8.66e15T^{2} \)
61 \( 1 + 2.01e8T + 1.16e16T^{2} \)
67 \( 1 + (-2.18e8 - 2.18e8i)T + 2.72e16iT^{2} \)
71 \( 1 + 8.88e7iT - 4.58e16T^{2} \)
73 \( 1 + (6.12e7 + 6.12e7i)T + 5.88e16iT^{2} \)
79 \( 1 + 3.41e8T + 1.19e17T^{2} \)
83 \( 1 + (-6.66e7 + 6.66e7i)T - 1.86e17iT^{2} \)
89 \( 1 - 1.49e8iT - 3.50e17T^{2} \)
97 \( 1 + (6.95e7 - 6.95e7i)T - 7.60e17iT^{2} \)
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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

−16.47467827467716106697295906096, −15.03302904264056024811128952263, −13.82686809146074835165069721314, −12.88405415233014505073867955890, −11.11101776910477530430280190357, −9.931359148084669724687720267867, −7.37113419334044857949819214433, −6.04673171649199664550158355565, −4.42339205291955579573170309045, −2.40121688456401633542086031856, 1.21279093618115065027501522066, 3.24656709472722276691652340517, 5.52926811341437718033364279389, 6.40825591096479747454268939574, 9.041009172419156761132340838555, 10.67120270526637654259585484950, 12.32628747408068303572563758592, 12.99141911675786264204197292464, 14.35998016507254579468725305891, 15.74969050103908813395143472259

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