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Properties

Label 2-20-5.3-c10-0-4
Degree $2$
Conductor $20$
Sign $-0.864 + 0.502i$
Analytic cond. $12.7071$
Root an. cond. $3.56470$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−190. − 190. i)3-s + (2.75e3 + 1.47e3i)5-s + (1.48e4 − 1.48e4i)7-s + 1.38e4i·9-s − 3.19e5·11-s + (−7.69e4 − 7.69e4i)13-s + (−2.44e5 − 8.07e5i)15-s + (−7.97e5 + 7.97e5i)17-s − 3.23e6i·19-s − 5.66e6·21-s + (−5.27e6 − 5.27e6i)23-s + (5.42e6 + 8.11e6i)25-s + (−8.63e6 + 8.63e6i)27-s − 2.58e7i·29-s + 1.45e7·31-s + ⋯
L(s)  = 1  + (−0.785 − 0.785i)3-s + (0.881 + 0.471i)5-s + (0.883 − 0.883i)7-s + 0.233i·9-s − 1.98·11-s + (−0.207 − 0.207i)13-s + (−0.322 − 1.06i)15-s + (−0.561 + 0.561i)17-s − 1.30i·19-s − 1.38·21-s + (−0.819 − 0.819i)23-s + (0.555 + 0.831i)25-s + (−0.601 + 0.601i)27-s − 1.25i·29-s + 0.508·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.864 + 0.502i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ -0.864 + 0.502i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.244889 - 0.908721i\)
\(L(\frac12)\) \(\approx\) \(0.244889 - 0.908721i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.75e3 - 1.47e3i)T \)
good3 \( 1 + (190. + 190. i)T + 5.90e4iT^{2} \)
7 \( 1 + (-1.48e4 + 1.48e4i)T - 2.82e8iT^{2} \)
11 \( 1 + 3.19e5T + 2.59e10T^{2} \)
13 \( 1 + (7.69e4 + 7.69e4i)T + 1.37e11iT^{2} \)
17 \( 1 + (7.97e5 - 7.97e5i)T - 2.01e12iT^{2} \)
19 \( 1 + 3.23e6iT - 6.13e12T^{2} \)
23 \( 1 + (5.27e6 + 5.27e6i)T + 4.14e13iT^{2} \)
29 \( 1 + 2.58e7iT - 4.20e14T^{2} \)
31 \( 1 - 1.45e7T + 8.19e14T^{2} \)
37 \( 1 + (2.45e7 - 2.45e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 6.80e7T + 1.34e16T^{2} \)
43 \( 1 + (5.22e6 + 5.22e6i)T + 2.16e16iT^{2} \)
47 \( 1 + (-4.11e7 + 4.11e7i)T - 5.25e16iT^{2} \)
53 \( 1 + (2.99e8 + 2.99e8i)T + 1.74e17iT^{2} \)
59 \( 1 - 4.68e7iT - 5.11e17T^{2} \)
61 \( 1 - 2.29e8T + 7.13e17T^{2} \)
67 \( 1 + (-1.40e9 + 1.40e9i)T - 1.82e18iT^{2} \)
71 \( 1 - 1.91e9T + 3.25e18T^{2} \)
73 \( 1 + (-2.58e9 - 2.58e9i)T + 4.29e18iT^{2} \)
79 \( 1 + 5.01e9iT - 9.46e18T^{2} \)
83 \( 1 + (-2.23e9 - 2.23e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 7.96e9iT - 3.11e19T^{2} \)
97 \( 1 + (4.52e9 - 4.52e9i)T - 7.37e19iT^{2} \)
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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

−15.36240464106389624025616074277, −13.75981559386201487692420295801, −12.91201371611757494969120369961, −11.17834438550338162280754347240, −10.26244167209522369150293630971, −7.895059124504254289687675532472, −6.55576650705049135492632154485, −5.07224218279528512065442912879, −2.21957458338221391986803129776, −0.41733292653824843150257176815, 2.12090337583122778846840120745, 4.98698903066272825885595275419, 5.58028559558017294842452669300, 8.177371246686362804235104961303, 9.831927219474265406094750433041, 10.91802044949414892123653055462, 12.36648755889847201153782319490, 13.86337601090972180245872693549, 15.49393354769510121580556863382, 16.37952507667391032939200304838

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