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Properties

Label 2-20-20.3-c9-0-1
Degree $2$
Conductor $20$
Sign $-0.876 + 0.481i$
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  + (12.1 + 19.1i)2-s + (48.3 − 48.3i)3-s + (−217. + 463. i)4-s + (−1.39e3 − 104. i)5-s + (1.50e3 + 337. i)6-s + (−5.61e3 − 5.61e3i)7-s + (−1.14e4 + 1.45e3i)8-s + 1.50e4i·9-s + (−1.48e4 − 2.78e4i)10-s + 5.94e4i·11-s + (1.18e4 + 3.29e4i)12-s + (−1.24e5 − 1.24e5i)13-s + (3.91e4 − 1.75e5i)14-s + (−7.24e4 + 6.23e4i)15-s + (−1.67e5 − 2.01e5i)16-s + (2.11e5 − 2.11e5i)17-s + ⋯
L(s)  = 1  + (0.535 + 0.844i)2-s + (0.344 − 0.344i)3-s + (−0.425 + 0.904i)4-s + (−0.997 − 0.0750i)5-s + (0.475 + 0.106i)6-s + (−0.883 − 0.883i)7-s + (−0.992 + 0.125i)8-s + 0.762i·9-s + (−0.471 − 0.882i)10-s + 1.22i·11-s + (0.165 + 0.458i)12-s + (−1.20 − 1.20i)13-s + (0.272 − 1.21i)14-s + (−0.369 + 0.317i)15-s + (−0.637 − 0.770i)16-s + (0.613 − 0.613i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.876 + 0.481i$
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: \((2,\ 20,\ (\ :9/2),\ -0.876 + 0.481i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.0978568 - 0.381366i\)
\(L(\frac12)\) \(\approx\) \(0.0978568 - 0.381366i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-12.1 - 19.1i)T \)
5 \( 1 + (1.39e3 + 104. i)T \)
good3 \( 1 + (-48.3 + 48.3i)T - 1.96e4iT^{2} \)
7 \( 1 + (5.61e3 + 5.61e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 5.94e4iT - 2.35e9T^{2} \)
13 \( 1 + (1.24e5 + 1.24e5i)T + 1.06e10iT^{2} \)
17 \( 1 + (-2.11e5 + 2.11e5i)T - 1.18e11iT^{2} \)
19 \( 1 - 2.38e5T + 3.22e11T^{2} \)
23 \( 1 + (1.77e6 - 1.77e6i)T - 1.80e12iT^{2} \)
29 \( 1 - 2.63e6iT - 1.45e13T^{2} \)
31 \( 1 - 2.40e6iT - 2.64e13T^{2} \)
37 \( 1 + (-2.79e6 + 2.79e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 4.05e6T + 3.27e14T^{2} \)
43 \( 1 + (1.38e7 - 1.38e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (-6.81e6 - 6.81e6i)T + 1.11e15iT^{2} \)
53 \( 1 + (4.59e7 + 4.59e7i)T + 3.29e15iT^{2} \)
59 \( 1 - 4.44e7T + 8.66e15T^{2} \)
61 \( 1 + 6.48e7T + 1.16e16T^{2} \)
67 \( 1 + (1.36e8 + 1.36e8i)T + 2.72e16iT^{2} \)
71 \( 1 - 3.40e7iT - 4.58e16T^{2} \)
73 \( 1 + (1.90e8 + 1.90e8i)T + 5.88e16iT^{2} \)
79 \( 1 - 2.42e7T + 1.19e17T^{2} \)
83 \( 1 + (1.78e8 - 1.78e8i)T - 1.86e17iT^{2} \)
89 \( 1 + 4.27e8iT - 3.50e17T^{2} \)
97 \( 1 + (9.14e7 - 9.14e7i)T - 7.60e17iT^{2} \)
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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

−16.59607189851806349282817638823, −15.64689024248359053918752959686, −14.40655316969092758867565840117, −13.10291486448895113360285338770, −12.15999212040817566310731347608, −9.914350132313026816368803494242, −7.72378068706695930055439286080, −7.29370705463164650946618680156, −4.92025305983772379573084644503, −3.25841305425836181603295597200, 0.14192069613427633967000722823, 2.85121794096002794198231383559, 4.08455580199033773837580113990, 6.19376897115440605477894538408, 8.722978960445673467161920946068, 9.927695118964076653619693294755, 11.73774283347469132319518287892, 12.37681312006172323437500477446, 14.19707115987655593301184690229, 15.18858461252637355179600176679

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