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Properties

Label 2-20-20.19-c10-0-8
Degree $2$
Conductor $20$
Sign $0.730 + 0.683i$
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  + (−0.302 − 31.9i)2-s − 375.·3-s + (−1.02e3 + 19.3i)4-s + (2.24e3 + 2.17e3i)5-s + (113. + 1.20e4i)6-s − 1.96e4·7-s + (929. + 3.27e4i)8-s + 8.21e4·9-s + (6.90e4 − 7.23e4i)10-s − 1.63e5i·11-s + (3.84e5 − 7.28e3i)12-s + 8.65e4i·13-s + (5.94e3 + 6.28e5i)14-s + (−8.42e5 − 8.18e5i)15-s + (1.04e6 − 3.96e4i)16-s − 2.09e6i·17-s + ⋯
L(s)  = 1  + (−0.00946 − 0.999i)2-s − 1.54·3-s + (−0.999 + 0.0189i)4-s + (0.717 + 0.696i)5-s + (0.0146 + 1.54i)6-s − 1.16·7-s + (0.0283 + 0.999i)8-s + 1.39·9-s + (0.690 − 0.723i)10-s − 1.01i·11-s + (1.54 − 0.0292i)12-s + 0.233i·13-s + (0.0110 + 1.16i)14-s + (−1.10 − 1.07i)15-s + (0.999 − 0.0378i)16-s − 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.730 + 0.683i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ 0.730 + 0.683i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.727981 - 0.287504i\)
\(L(\frac12)\) \(\approx\) \(0.727981 - 0.287504i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.302 + 31.9i)T \)
5 \( 1 + (-2.24e3 - 2.17e3i)T \)
good3 \( 1 + 375.T + 5.90e4T^{2} \)
7 \( 1 + 1.96e4T + 2.82e8T^{2} \)
11 \( 1 + 1.63e5iT - 2.59e10T^{2} \)
13 \( 1 - 8.65e4iT - 1.37e11T^{2} \)
17 \( 1 + 2.09e6iT - 2.01e12T^{2} \)
19 \( 1 - 3.13e6iT - 6.13e12T^{2} \)
23 \( 1 - 1.05e7T + 4.14e13T^{2} \)
29 \( 1 - 1.13e7T + 4.20e14T^{2} \)
31 \( 1 - 1.21e7iT - 8.19e14T^{2} \)
37 \( 1 - 4.87e7iT - 4.80e15T^{2} \)
41 \( 1 - 4.55e7T + 1.34e16T^{2} \)
43 \( 1 - 7.47e7T + 2.16e16T^{2} \)
47 \( 1 - 9.62e6T + 5.25e16T^{2} \)
53 \( 1 - 1.27e8iT - 1.74e17T^{2} \)
59 \( 1 + 1.18e9iT - 5.11e17T^{2} \)
61 \( 1 - 2.97e8T + 7.13e17T^{2} \)
67 \( 1 - 7.15e8T + 1.82e18T^{2} \)
71 \( 1 - 2.56e9iT - 3.25e18T^{2} \)
73 \( 1 + 2.10e9iT - 4.29e18T^{2} \)
79 \( 1 - 3.44e9iT - 9.46e18T^{2} \)
83 \( 1 - 7.09e9T + 1.55e19T^{2} \)
89 \( 1 + 3.39e9T + 3.11e19T^{2} \)
97 \( 1 - 5.71e9iT - 7.37e19T^{2} \)
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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

−16.24349929858015479346201360813, −14.00583648999173913315659827096, −12.80622740135669684283727352114, −11.53713056759804475035343054196, −10.57297141443198191680441625489, −9.485551716324171745526624685003, −6.56114089536192013922306358668, −5.33810763992005300513107878859, −3.09539081836977387504373645649, −0.78662815771096524444542795259, 0.68305808225644599227322633690, 4.67529783132593254545666275080, 5.88158181169071627625740601689, 6.86961416516294090670854487335, 9.193409347122246795414322859956, 10.39949663771264369543183770851, 12.57626751435729578243299925122, 13.10133164652595885921776213882, 15.24695600541656606326503511888, 16.38057886990808423545543013798

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