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Properties

Label 2-20-4.3-c8-0-13
Degree $2$
Conductor $20$
Sign $0.00639 + 0.999i$
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  + (11.3 + 11.2i)2-s − 137. i·3-s + (1.63 + 255. i)4-s − 279.·5-s + (1.54e3 − 1.55e3i)6-s − 3.94e3i·7-s + (−2.86e3 + 2.92e3i)8-s − 1.22e4·9-s + (−3.17e3 − 3.15e3i)10-s − 1.70e4i·11-s + (3.51e4 − 224. i)12-s + 1.09e4·13-s + (4.44e4 − 4.47e4i)14-s + 3.83e4i·15-s + (−6.55e4 + 837. i)16-s + 1.01e5·17-s + ⋯
L(s)  = 1  + (0.709 + 0.704i)2-s − 1.69i·3-s + (0.00639 + 0.999i)4-s − 0.447·5-s + (1.19 − 1.20i)6-s − 1.64i·7-s + (−0.700 + 0.713i)8-s − 1.87·9-s + (−0.317 − 0.315i)10-s − 1.16i·11-s + (1.69 − 0.0108i)12-s + 0.382·13-s + (1.15 − 1.16i)14-s + 0.758i·15-s + (−0.999 + 0.0127i)16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00639 + 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.00639 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.00639 + 0.999i$
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: \((2,\ 20,\ (\ :4),\ 0.00639 + 0.999i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.36748 - 1.35877i\)
\(L(\frac12)\) \(\approx\) \(1.36748 - 1.35877i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-11.3 - 11.2i)T \)
5 \( 1 + 279.T \)
good3 \( 1 + 137. iT - 6.56e3T^{2} \)
7 \( 1 + 3.94e3iT - 5.76e6T^{2} \)
11 \( 1 + 1.70e4iT - 2.14e8T^{2} \)
13 \( 1 - 1.09e4T + 8.15e8T^{2} \)
17 \( 1 - 1.01e5T + 6.97e9T^{2} \)
19 \( 1 - 9.34e4iT - 1.69e10T^{2} \)
23 \( 1 - 1.47e5iT - 7.83e10T^{2} \)
29 \( 1 - 4.16e4T + 5.00e11T^{2} \)
31 \( 1 + 1.38e5iT - 8.52e11T^{2} \)
37 \( 1 - 1.14e6T + 3.51e12T^{2} \)
41 \( 1 - 3.83e6T + 7.98e12T^{2} \)
43 \( 1 + 3.18e6iT - 1.16e13T^{2} \)
47 \( 1 - 3.51e6iT - 2.38e13T^{2} \)
53 \( 1 - 5.66e6T + 6.22e13T^{2} \)
59 \( 1 + 1.69e7iT - 1.46e14T^{2} \)
61 \( 1 + 5.16e6T + 1.91e14T^{2} \)
67 \( 1 + 1.05e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.85e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.38e6T + 8.06e14T^{2} \)
79 \( 1 - 4.42e7iT - 1.51e15T^{2} \)
83 \( 1 + 1.51e7iT - 2.25e15T^{2} \)
89 \( 1 + 5.42e7T + 3.93e15T^{2} \)
97 \( 1 + 1.24e8T + 7.83e15T^{2} \)
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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

−16.36540639430008475245707495887, −14.24201608504089471359122651454, −13.64925799657534531807690743186, −12.55463803055561451925361700975, −11.24158259295077524472241577551, −8.080815603836276848294537040692, −7.36684260914180765927520677340, −6.03245422712125512975480528175, −3.53993062341067221401711665139, −0.827552967984183215476600622154, 2.81296862561302362457495402456, 4.41877032469059534337251690107, 5.60643260250635362798199531104, 8.990526704207073714655236364856, 10.02406354813466651328578297950, 11.40956179390367966944606133711, 12.45413087797492407902785851676, 14.67677048026599577672773438059, 15.22491161414609018322068062044, 16.16050404038945246581141329432

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