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Properties

Label 1-4000-4000.563-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.708 + 0.705i$
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  + (0.940 + 0.338i)3-s + (−0.809 + 0.587i)7-s + (0.770 + 0.637i)9-s + (0.750 + 0.661i)11-s + (−0.0941 − 0.995i)13-s + (0.125 − 0.992i)17-s + (−0.940 + 0.338i)19-s + (−0.960 + 0.278i)21-s + (−0.929 + 0.368i)23-s + (0.509 + 0.860i)27-s + (0.999 + 0.0314i)29-s + (0.992 + 0.125i)31-s + (0.481 + 0.876i)33-s + (−0.860 − 0.509i)37-s + (0.248 − 0.968i)39-s + ⋯
L(s)  = 1  + (0.940 + 0.338i)3-s + (−0.809 + 0.587i)7-s + (0.770 + 0.637i)9-s + (0.750 + 0.661i)11-s + (−0.0941 − 0.995i)13-s + (0.125 − 0.992i)17-s + (−0.940 + 0.338i)19-s + (−0.960 + 0.278i)21-s + (−0.929 + 0.368i)23-s + (0.509 + 0.860i)27-s + (0.999 + 0.0314i)29-s + (0.992 + 0.125i)31-s + (0.481 + 0.876i)33-s + (−0.860 − 0.509i)37-s + (0.248 − 0.968i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.708 + 0.705i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.708 + 0.705i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.219878778 + 0.9174619366i\)
\(L(\frac12)\) \(\approx\) \(2.219878778 + 0.9174619366i\)
\(L(1)\) \(\approx\) \(1.401872141 + 0.2798652314i\)
\(L(1)\) \(\approx\) \(1.401872141 + 0.2798652314i\)

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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.750 + 0.661i)T \)
13 \( 1 + (-0.0941 - 0.995i)T \)
17 \( 1 + (0.125 - 0.992i)T \)
19 \( 1 + (-0.940 + 0.338i)T \)
23 \( 1 + (-0.929 + 0.368i)T \)
29 \( 1 + (0.999 + 0.0314i)T \)
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 \( 1 + (0.982 + 0.187i)T \)
53 \( 1 + (-0.278 - 0.960i)T \)
59 \( 1 + (0.975 + 0.218i)T \)
61 \( 1 + (0.917 + 0.397i)T \)
67 \( 1 + (0.999 - 0.0314i)T \)
71 \( 1 + (-0.982 - 0.187i)T \)
73 \( 1 + (-0.535 + 0.844i)T \)
79 \( 1 + (0.425 + 0.904i)T \)
83 \( 1 + (0.338 + 0.940i)T \)
89 \( 1 + (0.844 + 0.535i)T \)
97 \( 1 + (-0.684 - 0.728i)T \)
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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

−18.78219525626570567385932970272, −17.59721005058177560898121234675, −17.113460965199028820118363974970, −16.25870350937478450090299620461, −15.7502689047524153105680780326, −14.80583721028195767821911442541, −14.205703319688789793837590410190, −13.727537961572105323150632881498, −13.04543412587107917665606934647, −12.33610914191522827136270969037, −11.70364248020111682859888543797, −10.60575819632473918260973825622, −10.05950850119763432220236420585, −9.25964171182531095639079362302, −8.65106600512413630984195061250, −8.06190485466362750831621822935, −7.12251393141444021429053580140, −6.35865737991333119990518846304, −6.199515191781427338661227421013, −4.440248664499504578918122810912, −4.101448703020824689310680893752, −3.304711241399689519653070323695, −2.49975145873438280570562686207, −1.6315498508293426221481215934, −0.738449625881088229998400691270, 0.87360005420805073893595093847, 2.1999591964676618418800655051, 2.55437308494566191263005432924, 3.551163770477940521185465813357, 4.07758786127482338490613285787, 5.04072635739599366907933589469, 5.82999500392446203020912482366, 6.770222332748883020981109075815, 7.36327848472699549127082814211, 8.32792401508193166640301551626, 8.78973917514551769862221366062, 9.67227899310941121413376672294, 10.00396337113289052227188263672, 10.75631829823356674609028183049, 12.0133784043411883136343669374, 12.3687647646555359404109120387, 13.12759958811925703519803990006, 13.94125689195668691520199896945, 14.45040231407041817605515563351, 15.25979947957428570934685797709, 15.78374625270846133462937570478, 16.20376206398705130334314893925, 17.340714819655209051930331689614, 17.83037351750741760897547566546, 18.81665354185193388769988625146

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