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Properties

Label 2-20-4.3-c6-0-6
Degree $2$
Conductor $20$
Sign $0.870 - 0.492i$
Analytic cond. $4.60108$
Root an. cond. $2.14501$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.73 − 2.03i)2-s + 28.9i·3-s + (55.6 − 31.5i)4-s + 55.9·5-s + (59.0 + 224. i)6-s + 435. i·7-s + (366. − 357. i)8-s − 110.·9-s + (432. − 113. i)10-s − 2.21e3i·11-s + (913. + 1.61e3i)12-s − 1.93e3·13-s + (888. + 3.37e3i)14-s + 1.62e3i·15-s + (2.10e3 − 3.51e3i)16-s − 5.66e3·17-s + ⋯
L(s)  = 1  + (0.967 − 0.254i)2-s + 1.07i·3-s + (0.870 − 0.492i)4-s + 0.447·5-s + (0.273 + 1.03i)6-s + 1.27i·7-s + (0.715 − 0.698i)8-s − 0.152·9-s + (0.432 − 0.113i)10-s − 1.66i·11-s + (0.528 + 0.934i)12-s − 0.878·13-s + (0.323 + 1.22i)14-s + 0.480i·15-s + (0.514 − 0.857i)16-s − 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.870 - 0.492i$
Analytic conductor: \(4.60108\)
Root analytic conductor: \(2.14501\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3),\ 0.870 - 0.492i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.59093 + 0.682576i\)
\(L(\frac12)\) \(\approx\) \(2.59093 + 0.682576i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.73 + 2.03i)T \)
5 \( 1 - 55.9T \)
good3 \( 1 - 28.9iT - 729T^{2} \)
7 \( 1 - 435. iT - 1.17e5T^{2} \)
11 \( 1 + 2.21e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.93e3T + 4.82e6T^{2} \)
17 \( 1 + 5.66e3T + 2.41e7T^{2} \)
19 \( 1 + 1.61e3iT - 4.70e7T^{2} \)
23 \( 1 + 9.24e3iT - 1.48e8T^{2} \)
29 \( 1 - 2.17e4T + 5.94e8T^{2} \)
31 \( 1 + 5.26e3iT - 8.87e8T^{2} \)
37 \( 1 - 3.52e4T + 2.56e9T^{2} \)
41 \( 1 + 7.88e4T + 4.75e9T^{2} \)
43 \( 1 - 1.24e5iT - 6.32e9T^{2} \)
47 \( 1 + 1.81e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.31e5T + 2.21e10T^{2} \)
59 \( 1 - 2.81e5iT - 4.21e10T^{2} \)
61 \( 1 + 4.15e5T + 5.15e10T^{2} \)
67 \( 1 + 2.12e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.03e5iT - 1.28e11T^{2} \)
73 \( 1 + 5.01e5T + 1.51e11T^{2} \)
79 \( 1 + 8.07e4iT - 2.43e11T^{2} \)
83 \( 1 + 5.09e5iT - 3.26e11T^{2} \)
89 \( 1 - 7.77e5T + 4.96e11T^{2} \)
97 \( 1 - 5.99e5T + 8.32e11T^{2} \)
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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

−16.55743845138394473460863897942, −15.60100549931447142191537709211, −14.64486034125547048693301457746, −13.23521601149274365760961998066, −11.71759217159107789472811259741, −10.46817553886708054440725470184, −8.986415662171721935503250278014, −6.12064540372273594412267360399, −4.76635861114787571162373707802, −2.78244953141032677067327109040, 1.93246961143733263645378479198, 4.55275389460691813718539727961, 6.78132730588684860317096789332, 7.46689451421407764657753425776, 10.23328146898601364175278655544, 12.08451923710550250701854190116, 13.10041683011100163801555644499, 13.94932893423059939167627750393, 15.31598869502810397287713406959, 17.08156780531784733242012638765

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