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Properties

Label 2-320-20.3-c3-0-17
Degree $2$
Conductor $320$
Sign $0.913 - 0.406i$
Analytic cond. $18.8806$
Root an. cond. $4.34518$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.50 + 4.50i)3-s + (6.30 − 9.23i)5-s + (19.0 + 19.0i)7-s − 13.6i·9-s − 24.0i·11-s + (−35.4 − 35.4i)13-s + (13.2 + 70.0i)15-s + (51.7 − 51.7i)17-s + 68.8·19-s − 171.·21-s + (63.6 − 63.6i)23-s + (−45.5 − 116. i)25-s + (−60.3 − 60.3i)27-s + 276. i·29-s + 165. i·31-s + ⋯
L(s)  = 1  + (−0.867 + 0.867i)3-s + (0.563 − 0.826i)5-s + (1.02 + 1.02i)7-s − 0.503i·9-s − 0.659i·11-s + (−0.755 − 0.755i)13-s + (0.227 + 1.20i)15-s + (0.738 − 0.738i)17-s + 0.831·19-s − 1.78·21-s + (0.577 − 0.577i)23-s + (−0.364 − 0.931i)25-s + (−0.430 − 0.430i)27-s + 1.77i·29-s + 0.958i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.913 - 0.406i$
Analytic conductor: \(18.8806\)
Root analytic conductor: \(4.34518\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :3/2),\ 0.913 - 0.406i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.705693426\)
\(L(\frac12)\) \(\approx\) \(1.705693426\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.30 + 9.23i)T \)
good3 \( 1 + (4.50 - 4.50i)T - 27iT^{2} \)
7 \( 1 + (-19.0 - 19.0i)T + 343iT^{2} \)
11 \( 1 + 24.0iT - 1.33e3T^{2} \)
13 \( 1 + (35.4 + 35.4i)T + 2.19e3iT^{2} \)
17 \( 1 + (-51.7 + 51.7i)T - 4.91e3iT^{2} \)
19 \( 1 - 68.8T + 6.85e3T^{2} \)
23 \( 1 + (-63.6 + 63.6i)T - 1.21e4iT^{2} \)
29 \( 1 - 276. iT - 2.43e4T^{2} \)
31 \( 1 - 165. iT - 2.97e4T^{2} \)
37 \( 1 + (-245. + 245. i)T - 5.06e4iT^{2} \)
41 \( 1 - 255.T + 6.89e4T^{2} \)
43 \( 1 + (10.3 - 10.3i)T - 7.95e4iT^{2} \)
47 \( 1 + (-352. - 352. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-275. - 275. i)T + 1.48e5iT^{2} \)
59 \( 1 - 245.T + 2.05e5T^{2} \)
61 \( 1 - 1.64T + 2.26e5T^{2} \)
67 \( 1 + (-544. - 544. i)T + 3.00e5iT^{2} \)
71 \( 1 - 127. iT - 3.57e5T^{2} \)
73 \( 1 + (786. + 786. i)T + 3.89e5iT^{2} \)
79 \( 1 - 111.T + 4.93e5T^{2} \)
83 \( 1 + (650. - 650. i)T - 5.71e5iT^{2} \)
89 \( 1 + 467. iT - 7.04e5T^{2} \)
97 \( 1 + (-695. + 695. i)T - 9.12e5iT^{2} \)
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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

−11.22065203837828673437889606271, −10.42530998037637793392237868919, −9.400420524762099224830124375518, −8.681585686919549330501454418042, −7.54308621155251560065996525924, −5.66353019795318216887426214243, −5.39412379019940731483826976329, −4.61098757314837825748197202823, −2.72604492737869838093028411893, −0.960246755297020521937782363200, 1.01953024293096630154431601362, 2.15693127226471530534225131711, 4.05876493053585308572135264652, 5.34580348783526100045464559329, 6.35945339471144457643839123209, 7.32490505015519658442451269464, 7.71654149470095949922745422626, 9.595333374145627214716829384452, 10.28317966218143920886879997430, 11.47580235330703351817251333922

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