Properties

Label 1-6008-6008.4557-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.994 + 0.105i$
Analytic cond. $27.9010$
Root an. cond. $27.9010$
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  + (0.929 − 0.368i)3-s + (−0.535 + 0.844i)5-s + (−0.929 + 0.368i)7-s + (0.728 − 0.684i)9-s + (0.809 + 0.587i)11-s + (0.992 + 0.125i)13-s + (−0.187 + 0.982i)15-s + (0.0627 − 0.998i)17-s + (−0.876 + 0.481i)19-s + (−0.728 + 0.684i)21-s + (0.876 + 0.481i)23-s + (−0.425 − 0.904i)25-s + (0.425 − 0.904i)27-s + (−0.728 + 0.684i)29-s + (0.728 − 0.684i)31-s + ⋯
L(s)  = 1  + (0.929 − 0.368i)3-s + (−0.535 + 0.844i)5-s + (−0.929 + 0.368i)7-s + (0.728 − 0.684i)9-s + (0.809 + 0.587i)11-s + (0.992 + 0.125i)13-s + (−0.187 + 0.982i)15-s + (0.0627 − 0.998i)17-s + (−0.876 + 0.481i)19-s + (−0.728 + 0.684i)21-s + (0.876 + 0.481i)23-s + (−0.425 − 0.904i)25-s + (0.425 − 0.904i)27-s + (−0.728 + 0.684i)29-s + (0.728 − 0.684i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.994 + 0.105i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (4557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.994 + 0.105i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.364638957 + 0.1254246887i\)
\(L(\frac12)\) \(\approx\) \(2.364638957 + 0.1254246887i\)
\(L(1)\) \(\approx\) \(1.359486030 + 0.05889468143i\)
\(L(1)\) \(\approx\) \(1.359486030 + 0.05889468143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.929 - 0.368i)T \)
5 \( 1 + (-0.535 + 0.844i)T \)
7 \( 1 + (-0.929 + 0.368i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.992 + 0.125i)T \)
17 \( 1 + (0.0627 - 0.998i)T \)
19 \( 1 + (-0.876 + 0.481i)T \)
23 \( 1 + (0.876 + 0.481i)T \)
29 \( 1 + (-0.728 + 0.684i)T \)
31 \( 1 + (0.728 - 0.684i)T \)
37 \( 1 + (0.425 + 0.904i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.535 - 0.844i)T \)
47 \( 1 + (0.535 - 0.844i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.637 + 0.770i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.728 + 0.684i)T \)
71 \( 1 + (-0.187 - 0.982i)T \)
73 \( 1 + T \)
79 \( 1 + (0.876 + 0.481i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.535 - 0.844i)T \)
97 \( 1 + (-0.929 - 0.368i)T \)
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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

−17.514767451121818398744621078664, −16.71114488005845782579484470446, −16.482858377049281503687925246539, −15.683484581613753556560816491107, −15.191555794300552823411298813425, −14.50832539570370147401982807955, −13.624152198871493349646496484488, −13.08673975389858629617413584101, −12.78142374155559928363790736383, −11.80256262091884069724458470995, −10.906842175864896353400272583732, −10.47224353016741960689879954186, −9.38962876219479105610703399159, −9.10344464951084768483066060004, −8.34774269449272692917545574528, −7.99280129154726185466937746262, −6.86655738263389218801785529607, −6.35772891322312564828626090030, −5.423755821540018176350358137023, −4.362257210258451520433125939527, −3.960409503184641445616678427609, −3.4225923622700536683519276772, −2.59344195363392426877000531096, −1.463859826500663137025148771960, −0.79010627096484769337099992275, 0.71722841241656279570377493591, 1.837726684549533206939920442348, 2.48662480354815770221198421829, 3.436990149023501578541975724568, 3.600230370261420250037782798081, 4.49144981855115871535683622067, 5.71921034430567869607537035602, 6.60333743951877863280705320943, 6.89350492927047881562169998753, 7.51840844814329917336509832540, 8.55891909623016072076425090917, 8.87027479436917925886301111417, 9.77839516039181431759722717246, 10.1996847474021987973436553088, 11.24508231012990524616682230515, 11.881574173691444586603051529545, 12.43595826720664059535564370303, 13.38947592294328112686658284978, 13.62769614474672831220853663459, 14.60005910090853993026613806211, 15.12852543125445555693410253600, 15.46105747606385635878037213998, 16.32003432270302628378452046711, 17.007765346651547639810779526366, 18.047287370425667643323758781246

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