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Properties

Label 2-20-20.7-c5-0-0
Degree $2$
Conductor $20$
Sign $-0.819 + 0.572i$
Analytic cond. $3.20767$
Root an. cond. $1.79099$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.216 + 5.65i)2-s + (−10.7 − 10.7i)3-s + (−31.9 + 2.44i)4-s + (−55.8 − 1.12i)5-s + (58.4 − 63.1i)6-s + (−26.5 + 26.5i)7-s + (−20.6 − 179. i)8-s − 11.5i·9-s + (−5.70 − 316. i)10-s + 430. i·11-s + (369. + 316. i)12-s + (−557. + 557. i)13-s + (−155. − 144. i)14-s + (589. + 613. i)15-s + (1.01e3 − 155. i)16-s + (−780. − 780. i)17-s + ⋯
L(s)  = 1  + (0.0381 + 0.999i)2-s + (−0.690 − 0.690i)3-s + (−0.997 + 0.0763i)4-s + (−0.999 − 0.0201i)5-s + (0.663 − 0.716i)6-s + (−0.204 + 0.204i)7-s + (−0.114 − 0.993i)8-s − 0.0473i·9-s + (−0.0180 − 0.999i)10-s + 1.07i·11-s + (0.740 + 0.635i)12-s + (−0.915 + 0.915i)13-s + (−0.212 − 0.197i)14-s + (0.676 + 0.703i)15-s + (0.988 − 0.152i)16-s + (−0.655 − 0.655i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.819 + 0.572i$
Analytic conductor: \(3.20767\)
Root analytic conductor: \(1.79099\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5/2),\ -0.819 + 0.572i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0132892 - 0.0422266i\)
\(L(\frac12)\) \(\approx\) \(0.0132892 - 0.0422266i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.216 - 5.65i)T \)
5 \( 1 + (55.8 + 1.12i)T \)
good3 \( 1 + (10.7 + 10.7i)T + 243iT^{2} \)
7 \( 1 + (26.5 - 26.5i)T - 1.68e4iT^{2} \)
11 \( 1 - 430. iT - 1.61e5T^{2} \)
13 \( 1 + (557. - 557. i)T - 3.71e5iT^{2} \)
17 \( 1 + (780. + 780. i)T + 1.41e6iT^{2} \)
19 \( 1 - 1.41e3T + 2.47e6T^{2} \)
23 \( 1 + (3.11e3 + 3.11e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 1.87e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.92e3iT - 2.86e7T^{2} \)
37 \( 1 + (517. + 517. i)T + 6.93e7iT^{2} \)
41 \( 1 + 8.37e3T + 1.15e8T^{2} \)
43 \( 1 + (-2.30e3 - 2.30e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (1.70e4 - 1.70e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (1.22e3 - 1.22e3i)T - 4.18e8iT^{2} \)
59 \( 1 + 2.34e3T + 7.14e8T^{2} \)
61 \( 1 + 2.85e4T + 8.44e8T^{2} \)
67 \( 1 + (-2.46e4 + 2.46e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 5.49e4iT - 1.80e9T^{2} \)
73 \( 1 + (-3.02e4 + 3.02e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 3.81e4T + 3.07e9T^{2} \)
83 \( 1 + (3.02e4 + 3.02e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 1.23e5iT - 5.58e9T^{2} \)
97 \( 1 + (-3.00e3 - 3.00e3i)T + 8.58e9iT^{2} \)
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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

−17.94274205243622364937264144561, −16.72192556201017528974194202211, −15.61550614320732212681762182655, −14.41126910758633943765888221547, −12.63992972324493667293165418569, −11.82545465363900890094238189528, −9.450769924085633831685412177667, −7.58630771968001303428782230699, −6.59790207857919621188775565029, −4.58921694022114549600007449953, 0.03506169780123813818897657605, 3.59247256778565812764377882921, 5.23626600602487392635162820469, 8.126787613915817279865531189505, 10.02738560869880432616450583174, 11.08834674370133379074083191158, 12.06390799218070684110626234956, 13.63125012460065506893204274046, 15.36414893270743160056023845295, 16.56726986132932850372741104699

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