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Properties

Label 2-20-20.3-c5-0-3
Degree $2$
Conductor $20$
Sign $0.361 - 0.932i$
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  + (−5.45 + 1.48i)2-s + (3.65 − 3.65i)3-s + (27.6 − 16.1i)4-s + (−18.9 + 52.5i)5-s + (−14.5 + 25.3i)6-s + (162. + 162. i)7-s + (−126. + 129. i)8-s + 216. i·9-s + (25.7 − 315. i)10-s − 539. i·11-s + (41.8 − 160. i)12-s + (297. + 297. i)13-s + (−1.13e3 − 648. i)14-s + (122. + 261. i)15-s + (501. − 892. i)16-s + (−350. + 350. i)17-s + ⋯
L(s)  = 1  + (−0.965 + 0.261i)2-s + (0.234 − 0.234i)3-s + (0.863 − 0.505i)4-s + (−0.339 + 0.940i)5-s + (−0.165 + 0.287i)6-s + (1.25 + 1.25i)7-s + (−0.700 + 0.713i)8-s + 0.889i·9-s + (0.0814 − 0.996i)10-s − 1.34i·11-s + (0.0839 − 0.320i)12-s + (0.488 + 0.488i)13-s + (−1.54 − 0.884i)14-s + (0.141 + 0.300i)15-s + (0.489 − 0.871i)16-s + (−0.293 + 0.293i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.361 - 0.932i$
Analytic conductor: \(3.20767\)
Root analytic conductor: \(1.79099\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5/2),\ 0.361 - 0.932i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.829665 + 0.567888i\)
\(L(\frac12)\) \(\approx\) \(0.829665 + 0.567888i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.45 - 1.48i)T \)
5 \( 1 + (18.9 - 52.5i)T \)
good3 \( 1 + (-3.65 + 3.65i)T - 243iT^{2} \)
7 \( 1 + (-162. - 162. i)T + 1.68e4iT^{2} \)
11 \( 1 + 539. iT - 1.61e5T^{2} \)
13 \( 1 + (-297. - 297. i)T + 3.71e5iT^{2} \)
17 \( 1 + (350. - 350. i)T - 1.41e6iT^{2} \)
19 \( 1 + 220.T + 2.47e6T^{2} \)
23 \( 1 + (314. - 314. i)T - 6.43e6iT^{2} \)
29 \( 1 + 1.57e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.10e3iT - 2.86e7T^{2} \)
37 \( 1 + (-3.96e3 + 3.96e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 8.30e3T + 1.15e8T^{2} \)
43 \( 1 + (-8.38e3 + 8.38e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (3.66e3 + 3.66e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (657. + 657. i)T + 4.18e8iT^{2} \)
59 \( 1 - 2.99e4T + 7.14e8T^{2} \)
61 \( 1 - 2.20e4T + 8.44e8T^{2} \)
67 \( 1 + (2.90e4 + 2.90e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 3.94e4iT - 1.80e9T^{2} \)
73 \( 1 + (3.44e4 + 3.44e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 1.74e4T + 3.07e9T^{2} \)
83 \( 1 + (7.49e3 - 7.49e3i)T - 3.93e9iT^{2} \)
89 \( 1 - 6.07e4iT - 5.58e9T^{2} \)
97 \( 1 + (7.40e4 - 7.40e4i)T - 8.58e9iT^{2} \)
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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

−17.85641553777543844722039813139, −16.25405741588500963237923384240, −15.09325388391720766875130246817, −14.04387039620281849518631637614, −11.54420416186349848357380179026, −10.89000357815974607524934784984, −8.727216473105348867146382020190, −7.81367365436285104640636899527, −5.91432525089648554068990198733, −2.26099711757002984713237222359, 1.12671489703964996035028392121, 4.26834529643224497079728503954, 7.33003349708165727445600367166, 8.567335076918236624365247854116, 9.995398200979599892615337665436, 11.42501152920982009915483097529, 12.75232953415127353107661235948, 14.72830682465063910727411629641, 15.99167310503911369592003253744, 17.39629464036593457781967126022

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