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Properties

Label 1-4000-4000.1483-r0-0-0
Degree $1$
Conductor $4000$
Sign $-0.444 + 0.895i$
Analytic cond. $18.5759$
Root an. cond. $18.5759$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.995 − 0.0941i)3-s + (−0.309 + 0.951i)7-s + (0.982 − 0.187i)9-s + (0.790 + 0.612i)11-s + (−0.562 + 0.827i)13-s + (−0.248 + 0.968i)17-s + (0.995 + 0.0941i)19-s + (−0.218 + 0.975i)21-s + (−0.728 + 0.684i)23-s + (0.960 − 0.278i)27-s + (0.661 + 0.750i)29-s + (−0.968 − 0.248i)31-s + (0.844 + 0.535i)33-s + (−0.278 + 0.960i)37-s + (−0.481 + 0.876i)39-s + ⋯
L(s)  = 1  + (0.995 − 0.0941i)3-s + (−0.309 + 0.951i)7-s + (0.982 − 0.187i)9-s + (0.790 + 0.612i)11-s + (−0.562 + 0.827i)13-s + (−0.248 + 0.968i)17-s + (0.995 + 0.0941i)19-s + (−0.218 + 0.975i)21-s + (−0.728 + 0.684i)23-s + (0.960 − 0.278i)27-s + (0.661 + 0.750i)29-s + (−0.968 − 0.248i)31-s + (0.844 + 0.535i)33-s + (−0.278 + 0.960i)37-s + (−0.481 + 0.876i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ -0.444 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.150174040 + 1.853956894i\)
\(L(\frac12)\) \(\approx\) \(1.150174040 + 1.853956894i\)
\(L(1)\) \(\approx\) \(1.352711377 + 0.4483455194i\)
\(L(1)\) \(\approx\) \(1.352711377 + 0.4483455194i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.995 - 0.0941i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.790 + 0.612i)T \)
13 \( 1 + (-0.562 + 0.827i)T \)
17 \( 1 + (-0.248 + 0.968i)T \)
19 \( 1 + (0.995 + 0.0941i)T \)
23 \( 1 + (-0.728 + 0.684i)T \)
29 \( 1 + (0.661 + 0.750i)T \)
31 \( 1 + (-0.968 - 0.248i)T \)
37 \( 1 + (-0.278 + 0.960i)T \)
41 \( 1 + (0.684 - 0.728i)T \)
43 \( 1 + (-0.987 + 0.156i)T \)
47 \( 1 + (-0.368 + 0.929i)T \)
53 \( 1 + (-0.975 - 0.218i)T \)
59 \( 1 + (0.940 - 0.338i)T \)
61 \( 1 + (0.0314 - 0.999i)T \)
67 \( 1 + (-0.661 + 0.750i)T \)
71 \( 1 + (-0.368 + 0.929i)T \)
73 \( 1 + (-0.425 - 0.904i)T \)
79 \( 1 + (0.637 - 0.770i)T \)
83 \( 1 + (0.0941 - 0.995i)T \)
89 \( 1 + (-0.904 + 0.425i)T \)
97 \( 1 + (-0.998 - 0.0627i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.161384803056869040452054229175, −17.80569677194393516329592909366, −16.61151248325900955025630913759, −16.3211641765032302576157997621, −15.5379432214234154190724378735, −14.707653161980898126670955568061, −14.07466869372185757000444517731, −13.69864486865175224645373936737, −12.96385366627576521518377303180, −12.18604602685595275289560985396, −11.349369134115433128039033390, −10.48404187092732842462193448070, −9.84709503679213529510103403084, −9.33321094815168879516792613399, −8.49044767961084163877291048288, −7.7619221040420599059070547684, −7.17457645109336552276744908375, −6.51160313402757600809056506895, −5.43093506584668104835710491343, −4.54243431308498522865450427418, −3.8271415890086698634112816757, −3.1630865291987622886173210904, −2.49348324205539224600591846458, −1.36949762000072683500969946836, −0.49712350734616514306546982985, 1.57213665181231621451521648573, 1.831389210980230763470899412423, 2.86922078801484678540233379247, 3.563540889036109845472989207490, 4.32899846757151480592694102742, 5.15793182733482905287878426667, 6.1843069552109425889390784829, 6.8333873576846577652417695871, 7.547439407974796524210047379026, 8.374829378136783842870085331789, 9.0301416683660791846211146261, 9.60155549685577845944575833941, 10.05104762716866263893806186904, 11.28206620320778072102538541967, 12.05487398072767949621390919497, 12.490556245614102859673247063539, 13.25174912153173904630635375153, 14.12464428452142999479292928972, 14.58722812572892156591357564358, 15.171428954609836477248828828996, 15.91223447201647770776382731947, 16.467153659277635273383125846121, 17.58664612379137798488655697974, 17.992955333805859245146309502799, 19.06332409408635642576240458042

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