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Properties

Label 1-4000-4000.1253-r1-0-0
Degree $1$
Conductor $4000$
Sign $0.641 + 0.767i$
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  + (0.661 − 0.750i)3-s + (0.309 + 0.951i)7-s + (−0.125 − 0.992i)9-s + (0.940 − 0.338i)11-s + (−0.790 − 0.612i)13-s + (−0.770 − 0.637i)17-s + (0.661 + 0.750i)19-s + (0.917 + 0.397i)21-s + (0.876 − 0.481i)23-s + (−0.827 − 0.562i)27-s + (−0.218 − 0.975i)29-s + (−0.637 + 0.770i)31-s + (0.368 − 0.929i)33-s + (0.562 + 0.827i)37-s + (−0.982 + 0.187i)39-s + ⋯
L(s)  = 1  + (0.661 − 0.750i)3-s + (0.309 + 0.951i)7-s + (−0.125 − 0.992i)9-s + (0.940 − 0.338i)11-s + (−0.790 − 0.612i)13-s + (−0.770 − 0.637i)17-s + (0.661 + 0.750i)19-s + (0.917 + 0.397i)21-s + (0.876 − 0.481i)23-s + (−0.827 − 0.562i)27-s + (−0.218 − 0.975i)29-s + (−0.637 + 0.770i)31-s + (0.368 − 0.929i)33-s + (0.562 + 0.827i)37-s + (−0.982 + 0.187i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.641 + 0.767i$
Analytic conductor: \(429.859\)
Root analytic conductor: \(429.859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (1:\ ),\ 0.641 + 0.767i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.049153691 + 0.9583677370i\)
\(L(\frac12)\) \(\approx\) \(2.049153691 + 0.9583677370i\)
\(L(1)\) \(\approx\) \(1.297565114 - 0.1828460267i\)
\(L(1)\) \(\approx\) \(1.297565114 - 0.1828460267i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.661 - 0.750i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.940 - 0.338i)T \)
13 \( 1 + (-0.790 - 0.612i)T \)
17 \( 1 + (-0.770 - 0.637i)T \)
19 \( 1 + (0.661 + 0.750i)T \)
23 \( 1 + (0.876 - 0.481i)T \)
29 \( 1 + (-0.218 - 0.975i)T \)
31 \( 1 + (-0.637 + 0.770i)T \)
37 \( 1 + (0.562 + 0.827i)T \)
41 \( 1 + (0.481 - 0.876i)T \)
43 \( 1 + (-0.156 + 0.987i)T \)
47 \( 1 + (-0.248 + 0.968i)T \)
53 \( 1 + (-0.397 + 0.917i)T \)
59 \( 1 + (-0.999 - 0.0314i)T \)
61 \( 1 + (-0.278 + 0.960i)T \)
67 \( 1 + (-0.218 + 0.975i)T \)
71 \( 1 + (-0.248 + 0.968i)T \)
73 \( 1 + (0.728 + 0.684i)T \)
79 \( 1 + (0.0627 + 0.998i)T \)
83 \( 1 + (0.750 - 0.661i)T \)
89 \( 1 + (0.684 - 0.728i)T \)
97 \( 1 + (-0.844 - 0.535i)T \)
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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

−18.11009203949636103204196721449, −17.38222019767042119569876238244, −16.74493338217952324577862145356, −16.367737957885729869535544952071, −15.21749924091874940708655475625, −14.8985021444173094407745132042, −14.20905704484829641215696454831, −13.5914284719146068100842815930, −12.94895491792872256173479203763, −11.93673676267888212824858084254, −11.00957498936963447206543924592, −10.82758492044032145494004144774, −9.55446534116540121910430977398, −9.44680267786078047384255420725, −8.6298969803206603105496101182, −7.6197123599237125977339293903, −7.18220983799168351303635510671, −6.37790188827925526512005287764, −5.05754017424314025730544196577, −4.67917069096623394202018714218, −3.85534855008657684422585222830, −3.33280270678501007335577163832, −2.16709194460590162427806527299, −1.56101544897688984594082284082, −0.30990946705391421829238192552, 0.89392243542791076294896515984, 1.59531470585473503259884226590, 2.64119116558487711578488753052, 2.93993822232586628500243713854, 4.04544099073244893387374763886, 4.95256751933594804902747243002, 5.83868748156645058489379547411, 6.45717506890910708457446652199, 7.31837874458336096006621995148, 7.90868163473203277943077626760, 8.71694383135116047392957520810, 9.224030903997274370946882613167, 9.83100800091181425278318191968, 11.05358078398295092985333199591, 11.69596955763862746278295927112, 12.302218945724623089760381837512, 12.85360775002257390959673954339, 13.701027317841380380097736886404, 14.3954056440418624403179995385, 14.83827373164694720828826582137, 15.51355099651401043511035105652, 16.3376261051195574089387779692, 17.32938236756765979543963336423, 17.73375016760790946419211599544, 18.62179987597519682565886616118

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