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Properties

Label 1-4000-4000.1011-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.923 + 0.384i$
Analytic cond. $429.859$
Root an. cond. $429.859$
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  + (−0.940 + 0.338i)3-s + (0.587 − 0.809i)7-s + (0.770 − 0.637i)9-s + (−0.661 − 0.750i)11-s + (0.0941 − 0.995i)13-s + (0.992 − 0.125i)17-s + (−0.338 + 0.940i)19-s + (−0.278 + 0.960i)21-s + (−0.368 + 0.929i)23-s + (−0.509 + 0.860i)27-s + (−0.0314 − 0.999i)29-s + (0.992 − 0.125i)31-s + (0.876 + 0.481i)33-s + (0.860 − 0.509i)37-s + (0.248 + 0.968i)39-s + ⋯
L(s)  = 1  + (−0.940 + 0.338i)3-s + (0.587 − 0.809i)7-s + (0.770 − 0.637i)9-s + (−0.661 − 0.750i)11-s + (0.0941 − 0.995i)13-s + (0.992 − 0.125i)17-s + (−0.338 + 0.940i)19-s + (−0.278 + 0.960i)21-s + (−0.368 + 0.929i)23-s + (−0.509 + 0.860i)27-s + (−0.0314 − 0.999i)29-s + (0.992 − 0.125i)31-s + (0.876 + 0.481i)33-s + (0.860 − 0.509i)37-s + (0.248 + 0.968i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.923 + 0.384i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.923 + 0.384i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.574770086 + 0.3145272049i\)
\(L(\frac12)\) \(\approx\) \(1.574770086 + 0.3145272049i\)
\(L(1)\) \(\approx\) \(0.8777122807 - 0.04383785763i\)
\(L(1)\) \(\approx\) \(0.8777122807 - 0.04383785763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.940 + 0.338i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (-0.661 - 0.750i)T \)
13 \( 1 + (0.0941 - 0.995i)T \)
17 \( 1 + (0.992 - 0.125i)T \)
19 \( 1 + (-0.338 + 0.940i)T \)
23 \( 1 + (-0.368 + 0.929i)T \)
29 \( 1 + (-0.0314 - 0.999i)T \)
31 \( 1 + (0.992 - 0.125i)T \)
37 \( 1 + (0.860 - 0.509i)T \)
41 \( 1 + (-0.368 - 0.929i)T \)
43 \( 1 + (0.891 + 0.453i)T \)
47 \( 1 + (-0.187 - 0.982i)T \)
53 \( 1 + (-0.278 + 0.960i)T \)
59 \( 1 + (0.218 + 0.975i)T \)
61 \( 1 + (0.397 + 0.917i)T \)
67 \( 1 + (0.999 + 0.0314i)T \)
71 \( 1 + (0.982 - 0.187i)T \)
73 \( 1 + (-0.844 + 0.535i)T \)
79 \( 1 + (-0.425 + 0.904i)T \)
83 \( 1 + (-0.338 + 0.940i)T \)
89 \( 1 + (0.844 - 0.535i)T \)
97 \( 1 + (0.728 + 0.684i)T \)
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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

−18.339530140280231058926809315101, −17.558412824906668169583880767758, −17.105810478985207067955278693327, −16.129584598082271391180415253601, −15.810997305719501633968782137905, −14.80366509940642054914219369011, −14.31674188815224835804757988945, −13.29220990836162302243891460900, −12.6050605261099014244841191594, −12.13528622721764849113932762062, −11.42351948088259636658091792577, −10.8704339131492023590712144548, −10.02645899047309934583125810897, −9.36533425816244753952561730457, −8.33472747580161358206967210456, −7.816873508141713904819280764530, −6.847252089536457245562983622, −6.36501322913339556453995041789, −5.46642810522495842011279511956, −4.78730730995921132840956101279, −4.38557180422688933105142096615, −2.94703494233954564625220930401, −2.11837657564180720705803799115, −1.464871680439369003643469871874, −0.40185981554474356243356399982, 0.7206569943326865391592324333, 1.09223918847519511142588359502, 2.40224593803402589721394222947, 3.54560089093209995012435587628, 4.02289032600147462304569334693, 4.98620903639022905595587862170, 5.70421180551163289581691547159, 6.03313590009587522631424609040, 7.262654307024575272676773835895, 7.81040054493919238532214635987, 8.41015485704095553195150193896, 9.70033139780668021415671724055, 10.20743878275018925808498585527, 10.706373275690986109870852642860, 11.430862135701670232914309480573, 12.03754855742646570378228275642, 12.85611637055648190286125486902, 13.54214244244510481382601186921, 14.245260756048288821281715114900, 15.10703142986228721057281438101, 15.74303305836885472828300230169, 16.39421232922915884492217178522, 17.07346497202697410728721421892, 17.50566618740387209507553216600, 18.28757200350648763249498060244

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