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Properties

Label 2-20-4.3-c8-0-6
Degree $2$
Conductor $20$
Sign $-0.0453 - 0.998i$
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  + (11.5 + 11.0i)2-s − 27.2i·3-s + (11.6 + 255. i)4-s + 279.·5-s + (301. − 315. i)6-s + 3.32e3i·7-s + (−2.69e3 + 3.08e3i)8-s + 5.81e3·9-s + (3.23e3 + 3.08e3i)10-s + 6.36e3i·11-s + (6.96e3 − 316. i)12-s − 3.07e4·13-s + (−3.67e4 + 3.84e4i)14-s − 7.61e3i·15-s + (−6.52e4 + 5.94e3i)16-s + 1.22e5·17-s + ⋯
L(s)  = 1  + (0.722 + 0.690i)2-s − 0.336i·3-s + (0.0453 + 0.998i)4-s + 0.447·5-s + (0.232 − 0.243i)6-s + 1.38i·7-s + (−0.657 + 0.753i)8-s + 0.886·9-s + (0.323 + 0.308i)10-s + 0.434i·11-s + (0.335 − 0.0152i)12-s − 1.07·13-s + (−0.956 + 1.00i)14-s − 0.150i·15-s + (−0.995 + 0.0906i)16-s + 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0453 - 0.998i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0453 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.0453 - 0.998i$
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: \((2,\ 20,\ (\ :4),\ -0.0453 - 0.998i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.76013 + 1.84192i\)
\(L(\frac12)\) \(\approx\) \(1.76013 + 1.84192i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-11.5 - 11.0i)T \)
5 \( 1 - 279.T \)
good3 \( 1 + 27.2iT - 6.56e3T^{2} \)
7 \( 1 - 3.32e3iT - 5.76e6T^{2} \)
11 \( 1 - 6.36e3iT - 2.14e8T^{2} \)
13 \( 1 + 3.07e4T + 8.15e8T^{2} \)
17 \( 1 - 1.22e5T + 6.97e9T^{2} \)
19 \( 1 + 7.55e3iT - 1.69e10T^{2} \)
23 \( 1 + 4.06e5iT - 7.83e10T^{2} \)
29 \( 1 - 8.28e5T + 5.00e11T^{2} \)
31 \( 1 + 1.39e6iT - 8.52e11T^{2} \)
37 \( 1 + 3.86e5T + 3.51e12T^{2} \)
41 \( 1 + 9.72e5T + 7.98e12T^{2} \)
43 \( 1 - 4.61e6iT - 1.16e13T^{2} \)
47 \( 1 - 1.87e6iT - 2.38e13T^{2} \)
53 \( 1 + 8.01e6T + 6.22e13T^{2} \)
59 \( 1 + 6.18e6iT - 1.46e14T^{2} \)
61 \( 1 - 9.79e6T + 1.91e14T^{2} \)
67 \( 1 + 1.19e6iT - 4.06e14T^{2} \)
71 \( 1 + 3.62e7iT - 6.45e14T^{2} \)
73 \( 1 - 3.35e7T + 8.06e14T^{2} \)
79 \( 1 - 1.22e6iT - 1.51e15T^{2} \)
83 \( 1 - 6.96e7iT - 2.25e15T^{2} \)
89 \( 1 - 5.58e7T + 3.93e15T^{2} \)
97 \( 1 + 5.97e7T + 7.83e15T^{2} \)
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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

−16.59582933084759986609906663028, −15.29960908856120759017094948649, −14.36409701347149627024797133412, −12.71437335143860576082752903379, −12.13792840039791753533907761332, −9.664964288615806391987225601135, −7.924715907756776795450277283158, −6.37327611663984184992899119105, −4.90812343678034081120469374168, −2.46627143576614737631621513591, 1.19896164369901919403162380118, 3.54971512733925353034626238994, 5.08710260547899128537158860805, 7.12068369800270470224626270960, 9.830559145576479877759952417623, 10.48164564286974746569782655529, 12.20696732386647278446853318717, 13.54872555906903037057591399573, 14.42586648510647790836277293593, 15.98275853819857205879243724073

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