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Properties

Label 2-20-20.7-c7-0-3
Degree $2$
Conductor $20$
Sign $-0.386 - 0.922i$
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  + (−9.35 − 6.36i)2-s + (56.5 + 56.5i)3-s + (47.0 + 119. i)4-s + (−240. − 141. i)5-s + (−169. − 888. i)6-s + (−764. + 764. i)7-s + (317. − 1.41e3i)8-s + 4.20e3i·9-s + (1.34e3 + 2.85e3i)10-s + 1.50e3i·11-s + (−4.07e3 + 9.38e3i)12-s + (−5.39e3 + 5.39e3i)13-s + (1.20e4 − 2.28e3i)14-s + (−5.59e3 − 2.16e4i)15-s + (−1.19e4 + 1.11e4i)16-s + (2.87e3 + 2.87e3i)17-s + ⋯
L(s)  = 1  + (−0.826 − 0.562i)2-s + (1.20 + 1.20i)3-s + (0.367 + 0.930i)4-s + (−0.861 − 0.507i)5-s + (−0.319 − 1.67i)6-s + (−0.842 + 0.842i)7-s + (0.219 − 0.975i)8-s + 1.92i·9-s + (0.426 + 0.904i)10-s + 0.341i·11-s + (−0.680 + 1.56i)12-s + (−0.680 + 0.680i)13-s + (1.17 − 0.222i)14-s + (−0.427 − 1.65i)15-s + (−0.730 + 0.683i)16-s + (0.141 + 0.141i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.386 - 0.922i$
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: \((2,\ 20,\ (\ :7/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.544925 + 0.819598i\)
\(L(\frac12)\) \(\approx\) \(0.544925 + 0.819598i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (9.35 + 6.36i)T \)
5 \( 1 + (240. + 141. i)T \)
good3 \( 1 + (-56.5 - 56.5i)T + 2.18e3iT^{2} \)
7 \( 1 + (764. - 764. i)T - 8.23e5iT^{2} \)
11 \( 1 - 1.50e3iT - 1.94e7T^{2} \)
13 \( 1 + (5.39e3 - 5.39e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (-2.87e3 - 2.87e3i)T + 4.10e8iT^{2} \)
19 \( 1 - 1.54e3T + 8.93e8T^{2} \)
23 \( 1 + (-4.93e4 - 4.93e4i)T + 3.40e9iT^{2} \)
29 \( 1 + 8.40e4iT - 1.72e10T^{2} \)
31 \( 1 + 3.03e5iT - 2.75e10T^{2} \)
37 \( 1 + (-1.74e5 - 1.74e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 1.14e5T + 1.94e11T^{2} \)
43 \( 1 + (-5.85e5 - 5.85e5i)T + 2.71e11iT^{2} \)
47 \( 1 + (1.61e5 - 1.61e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (-2.22e5 + 2.22e5i)T - 1.17e12iT^{2} \)
59 \( 1 - 2.00e6T + 2.48e12T^{2} \)
61 \( 1 + 2.52e6T + 3.14e12T^{2} \)
67 \( 1 + (-2.91e5 + 2.91e5i)T - 6.06e12iT^{2} \)
71 \( 1 - 1.18e6iT - 9.09e12T^{2} \)
73 \( 1 + (3.12e6 - 3.12e6i)T - 1.10e13iT^{2} \)
79 \( 1 + 8.20e4T + 1.92e13T^{2} \)
83 \( 1 + (-4.66e5 - 4.66e5i)T + 2.71e13iT^{2} \)
89 \( 1 - 2.67e6iT - 4.42e13T^{2} \)
97 \( 1 + (6.84e6 + 6.84e6i)T + 8.07e13iT^{2} \)
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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

−16.82661540707266288248559595004, −15.83309800167615201306764910035, −15.03582784495989413811127629136, −12.96830678536332441110202391815, −11.53064043716542384294590643967, −9.699734591484959143190220635035, −9.122422889778429166782493180178, −7.78581276143234443925232824993, −4.17682090486187201614599319576, −2.73497133177513891218930714209, 0.63025723691239378952633759034, 2.98181263512263279190403444763, 6.83440058791399516140319917221, 7.52291265351994007050009218809, 8.794680361859627899587995575278, 10.51198573310825801387509758881, 12.52814042831993444132396126946, 13.99702005377954276892078139721, 14.93528941687157802035567181662, 16.31520619857249044208784131615

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