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Properties

Label 2-20-20.3-c7-0-14
Degree $2$
Conductor $20$
Sign $0.610 + 0.792i$
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  + (10.6 − 3.68i)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 + (787. − 1.21e3i)8-s + 53.0i·9-s + (2.43e3 + 2.01e3i)10-s − 705. i·11-s + (716. − 5.86e3i)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  + (0.945 − 0.325i)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 + (0.543 − 0.839i)8-s + 0.0242i·9-s + (0.769 + 0.638i)10-s − 0.159i·11-s + (0.119 − 0.980i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.610 + 0.792i$
Analytic conductor: \(6.24770\)
Root analytic conductor: \(2.49954\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :7/2),\ 0.610 + 0.792i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.96089 - 1.45628i\)
\(L(\frac12)\) \(\approx\) \(2.96089 - 1.45628i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-10.6 + 3.68i)T \)
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

−16.24876326420380725794863576835, −14.73566029464289142229706894946, −13.63368892687163672938042793311, −13.15043951965128742511256490215, −11.22621271709116621003621996705, −9.895494057281987716938187841821, −7.42699001691129450749128388555, −6.18560898882001045054454660068, −3.56073659827577827460639703631, −1.97400703777410224940056393839, 2.77209792796289808384052109478, 4.57623331205181207861829187042, 6.21830377667171317863274239592, 8.526582353660877580834840985084, 9.763080877684104961551567847472, 11.98606585600156243874456270131, 13.15001083214397266657502829062, 14.32129034387257545845009195316, 15.68665808811207052583773513915, 16.22926330806758803447052665877

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