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Properties

Label 2-20-20.3-c5-0-4
Degree $2$
Conductor $20$
Sign $0.148 - 0.988i$
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  + (5.64 − 0.294i)2-s + (−20.2 + 20.2i)3-s + (31.8 − 3.32i)4-s + (16.5 + 53.4i)5-s + (−108. + 120. i)6-s + (9.02 + 9.02i)7-s + (178. − 28.1i)8-s − 574. i·9-s + (109. + 296. i)10-s − 200. i·11-s + (−576. + 710. i)12-s + (362. + 362. i)13-s + (53.6 + 48.3i)14-s + (−1.41e3 − 745. i)15-s + (1.00e3 − 211. i)16-s + (465. − 465. i)17-s + ⋯
L(s)  = 1  + (0.998 − 0.0520i)2-s + (−1.29 + 1.29i)3-s + (0.994 − 0.104i)4-s + (0.295 + 0.955i)5-s + (−1.22 + 1.36i)6-s + (0.0695 + 0.0695i)7-s + (0.987 − 0.155i)8-s − 2.36i·9-s + (0.345 + 0.938i)10-s − 0.500i·11-s + (−1.15 + 1.42i)12-s + (0.595 + 0.595i)13-s + (0.0731 + 0.0658i)14-s + (−1.62 − 0.855i)15-s + (0.978 − 0.206i)16-s + (0.391 − 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.148 - 0.988i$
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: \((2,\ 20,\ (\ :5/2),\ 0.148 - 0.988i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.35184 + 1.16395i\)
\(L(\frac12)\) \(\approx\) \(1.35184 + 1.16395i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.64 + 0.294i)T \)
5 \( 1 + (-16.5 - 53.4i)T \)
good3 \( 1 + (20.2 - 20.2i)T - 243iT^{2} \)
7 \( 1 + (-9.02 - 9.02i)T + 1.68e4iT^{2} \)
11 \( 1 + 200. iT - 1.61e5T^{2} \)
13 \( 1 + (-362. - 362. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-465. + 465. i)T - 1.41e6iT^{2} \)
19 \( 1 - 582.T + 2.47e6T^{2} \)
23 \( 1 + (285. - 285. i)T - 6.43e6iT^{2} \)
29 \( 1 + 1.27e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.26e3iT - 2.86e7T^{2} \)
37 \( 1 + (6.27e3 - 6.27e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 1.16e4T + 1.15e8T^{2} \)
43 \( 1 + (-1.19e4 + 1.19e4i)T - 1.47e8iT^{2} \)
47 \( 1 + (4.13e3 + 4.13e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (1.37e4 + 1.37e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 3.81e4T + 7.14e8T^{2} \)
61 \( 1 - 3.00e4T + 8.44e8T^{2} \)
67 \( 1 + (3.07e3 + 3.07e3i)T + 1.35e9iT^{2} \)
71 \( 1 - 4.94e4iT - 1.80e9T^{2} \)
73 \( 1 + (-2.54e4 - 2.54e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 9.90e4T + 3.07e9T^{2} \)
83 \( 1 + (3.98e4 - 3.98e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 6.51e4iT - 5.58e9T^{2} \)
97 \( 1 + (4.46e4 - 4.46e4i)T - 8.58e9iT^{2} \)
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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

−17.14566458657299701233652585242, −16.07087669544534784121465565853, −15.14596668108327034953357550140, −13.86160938705228380713483928677, −11.81224617897902553294314351944, −11.05307384530100316877085757151, −9.943528708825998133773606446931, −6.55979766933551388366057825302, −5.39460481386876125967999445822, −3.66829310731881366334750895765, 1.41544671053104563461674453739, 5.07158269768732575135225895777, 6.20649209077934850833552959765, 7.75106745869622577388887822310, 10.77106289900906988519373566253, 12.18599576570234483797703823406, 12.75850998728749872964207602160, 13.85290379660974737897865731306, 15.93899668108470059973037613185, 16.99534606732205062320550190970

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