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Properties

Label 2-20-20.7-c7-0-14
Degree $2$
Conductor $20$
Sign $0.621 + 0.783i$
Analytic cond. $6.24770$
Root an. cond. $2.49954$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.68 + 10.6i)2-s + (−32.6 − 32.6i)3-s + (−100. + 78.8i)4-s + (145. − 238. i)5-s + (228. − 469. i)6-s + (669. − 669. i)7-s + (−1.21e3 − 787. i)8-s − 53.0i·9-s + (3.08e3 + 672. i)10-s − 705. i·11-s + (5.86e3 + 716. i)12-s + (1.97e3 − 1.97e3i)13-s + (9.63e3 + 4.69e3i)14-s + (−1.25e4 + 3.05e3i)15-s + (3.94e3 − 1.59e4i)16-s + (−1.96e4 − 1.96e4i)17-s + ⋯
L(s)  = 1  + (0.325 + 0.945i)2-s + (−0.698 − 0.698i)3-s + (−0.787 + 0.616i)4-s + (0.519 − 0.854i)5-s + (0.432 − 0.887i)6-s + (0.738 − 0.738i)7-s + (−0.839 − 0.543i)8-s − 0.0242i·9-s + (0.977 + 0.212i)10-s − 0.159i·11-s + (0.980 + 0.119i)12-s + (0.248 − 0.248i)13-s + (0.938 + 0.457i)14-s + (−0.959 + 0.233i)15-s + (0.240 − 0.970i)16-s + (−0.967 − 0.967i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.621 + 0.783i$
Analytic conductor: \(6.24770\)
Root analytic conductor: \(2.49954\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :7/2),\ 0.621 + 0.783i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.21098 - 0.584719i\)
\(L(\frac12)\) \(\approx\) \(1.21098 - 0.584719i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.68 - 10.6i)T \)
5 \( 1 + (-145. + 238. i)T \)
good3 \( 1 + (32.6 + 32.6i)T + 2.18e3iT^{2} \)
7 \( 1 + (-669. + 669. i)T - 8.23e5iT^{2} \)
11 \( 1 + 705. iT - 1.94e7T^{2} \)
13 \( 1 + (-1.97e3 + 1.97e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (1.96e4 + 1.96e4i)T + 4.10e8iT^{2} \)
19 \( 1 + 2.33e4T + 8.93e8T^{2} \)
23 \( 1 + (-7.37e4 - 7.37e4i)T + 3.40e9iT^{2} \)
29 \( 1 + 4.30e4iT - 1.72e10T^{2} \)
31 \( 1 + 1.47e5iT - 2.75e10T^{2} \)
37 \( 1 + (-4.06e5 - 4.06e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 1.79e5T + 1.94e11T^{2} \)
43 \( 1 + (-3.17e5 - 3.17e5i)T + 2.71e11iT^{2} \)
47 \( 1 + (-8.86e5 + 8.86e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (3.83e5 - 3.83e5i)T - 1.17e12iT^{2} \)
59 \( 1 + 2.21e6T + 2.48e12T^{2} \)
61 \( 1 + 1.07e6T + 3.14e12T^{2} \)
67 \( 1 + (-1.24e6 + 1.24e6i)T - 6.06e12iT^{2} \)
71 \( 1 + 6.52e5iT - 9.09e12T^{2} \)
73 \( 1 + (1.61e6 - 1.61e6i)T - 1.10e13iT^{2} \)
79 \( 1 - 2.74e6T + 1.92e13T^{2} \)
83 \( 1 + (2.04e6 + 2.04e6i)T + 2.71e13iT^{2} \)
89 \( 1 + 7.28e6iT - 4.42e13T^{2} \)
97 \( 1 + (-8.26e6 - 8.26e6i)T + 8.07e13iT^{2} \)
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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

−16.93189681711439204753735171454, −15.33500264350922788992426408082, −13.69046266070102769339472310829, −12.95247693068733101732364762546, −11.45483600659349045358326079764, −9.155754443861514587276027819992, −7.55641869951071403475800496863, −6.10150190864019003504570598939, −4.66159915008251487006719986965, −0.790358653029094539496503923608, 2.24372660914428720419494763055, 4.53176645265325673594697475856, 6.00013848246697235625811118672, 8.960538097473125006325037140560, 10.67476833677420889428394466245, 11.05548274298887605200006077749, 12.72177048217674093736512375669, 14.33867968670628684102868725295, 15.29813127921107969512669254829, 17.22586861263324077429337348065

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