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Properties

Label 2-20-20.3-c9-0-3
Degree $2$
Conductor $20$
Sign $-0.214 + 0.976i$
Analytic cond. $10.3007$
Root an. cond. $3.20947$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.62 + 22.1i)2-s + (−107. + 107. i)3-s + (−469. + 204. i)4-s + (−103. + 1.39e3i)5-s + (−2.88e3 − 1.88e3i)6-s + (1.03e3 + 1.03e3i)7-s + (−6.71e3 − 9.44e3i)8-s − 3.49e3i·9-s + (−3.13e4 + 4.15e3i)10-s − 5.58e4i·11-s + (2.84e4 − 7.25e4i)12-s + (7.70e4 + 7.70e4i)13-s + (−1.81e4 + 2.76e4i)14-s + (−1.38e5 − 1.61e5i)15-s + (1.78e5 − 1.92e5i)16-s + (1.32e5 − 1.32e5i)17-s + ⋯
L(s)  = 1  + (0.204 + 0.978i)2-s + (−0.767 + 0.767i)3-s + (−0.916 + 0.400i)4-s + (−0.0740 + 0.997i)5-s + (−0.907 − 0.594i)6-s + (0.162 + 0.162i)7-s + (−0.579 − 0.815i)8-s − 0.177i·9-s + (−0.991 + 0.131i)10-s − 1.15i·11-s + (0.395 − 1.01i)12-s + (0.748 + 0.748i)13-s + (−0.125 + 0.192i)14-s + (−0.708 − 0.821i)15-s + (0.679 − 0.733i)16-s + (0.384 − 0.384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.214 + 0.976i$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :9/2),\ -0.214 + 0.976i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.429947 - 0.534399i\)
\(L(\frac12)\) \(\approx\) \(0.429947 - 0.534399i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.62 - 22.1i)T \)
5 \( 1 + (103. - 1.39e3i)T \)
good3 \( 1 + (107. - 107. i)T - 1.96e4iT^{2} \)
7 \( 1 + (-1.03e3 - 1.03e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 5.58e4iT - 2.35e9T^{2} \)
13 \( 1 + (-7.70e4 - 7.70e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-1.32e5 + 1.32e5i)T - 1.18e11iT^{2} \)
19 \( 1 + 6.83e5T + 3.22e11T^{2} \)
23 \( 1 + (4.98e5 - 4.98e5i)T - 1.80e12iT^{2} \)
29 \( 1 - 6.68e6iT - 1.45e13T^{2} \)
31 \( 1 - 3.14e6iT - 2.64e13T^{2} \)
37 \( 1 + (-3.82e6 + 3.82e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 2.79e7T + 3.27e14T^{2} \)
43 \( 1 + (-6.41e6 + 6.41e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (3.41e7 + 3.41e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (-4.49e7 - 4.49e7i)T + 3.29e15iT^{2} \)
59 \( 1 - 1.33e8T + 8.66e15T^{2} \)
61 \( 1 + 4.19e7T + 1.16e16T^{2} \)
67 \( 1 + (-6.05e7 - 6.05e7i)T + 2.72e16iT^{2} \)
71 \( 1 - 2.54e8iT - 4.58e16T^{2} \)
73 \( 1 + (2.12e8 + 2.12e8i)T + 5.88e16iT^{2} \)
79 \( 1 + 3.42e8T + 1.19e17T^{2} \)
83 \( 1 + (9.07e7 - 9.07e7i)T - 1.86e17iT^{2} \)
89 \( 1 - 1.04e9iT - 3.50e17T^{2} \)
97 \( 1 + (1.03e8 - 1.03e8i)T - 7.60e17iT^{2} \)
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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

−16.69494761578036981863089062990, −15.99070134023964556411738411781, −14.73365880176990307458286210297, −13.62336333446650330772689219632, −11.56660370771703534608629848980, −10.38624710388005197173048481235, −8.577915478040553553085691435371, −6.70789870572640706518138773951, −5.45611999053427469794954266283, −3.72331618303463656362755437443, 0.34026533703511495350698302584, 1.66139143881367792180209236551, 4.33234559081023634402074552496, 5.93781287000189934518378131868, 8.218552146844552680563082842140, 9.949892340096275050011332253025, 11.52237221963055459863136743470, 12.56225636699588735772250260343, 13.20605181658545400868026638736, 15.10324232777365256678734185071

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