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Properties

Label 1-4001-4001.468-r0-0-0
Degree $1$
Conductor $4001$
Sign $-0.924 + 0.381i$
Analytic cond. $18.5805$
Root an. cond. $18.5805$
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  + (0.951 + 0.309i)2-s + (−0.852 − 0.522i)3-s + (0.809 + 0.587i)4-s + i·5-s + (−0.649 − 0.760i)6-s + (0.951 − 0.309i)7-s + (0.587 + 0.809i)8-s + (0.453 + 0.891i)9-s + (−0.309 + 0.951i)10-s + (0.382 + 0.923i)11-s + (−0.382 − 0.923i)12-s + (−0.587 + 0.809i)13-s + 14-s + (0.522 − 0.852i)15-s + (0.309 + 0.951i)16-s + (−0.233 − 0.972i)17-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (−0.852 − 0.522i)3-s + (0.809 + 0.587i)4-s + i·5-s + (−0.649 − 0.760i)6-s + (0.951 − 0.309i)7-s + (0.587 + 0.809i)8-s + (0.453 + 0.891i)9-s + (−0.309 + 0.951i)10-s + (0.382 + 0.923i)11-s + (−0.382 − 0.923i)12-s + (−0.587 + 0.809i)13-s + 14-s + (0.522 − 0.852i)15-s + (0.309 + 0.951i)16-s + (−0.233 − 0.972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4001\)
Sign: $-0.924 + 0.381i$
Analytic conductor: \(18.5805\)
Root analytic conductor: \(18.5805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4001} (468, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4001,\ (0:\ ),\ -0.924 + 0.381i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4084836937 + 2.059460012i\)
\(L(\frac12)\) \(\approx\) \(0.4084836937 + 2.059460012i\)
\(L(1)\) \(\approx\) \(1.295042257 + 0.7063033320i\)
\(L(1)\) \(\approx\) \(1.295042257 + 0.7063033320i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (-0.852 - 0.522i)T \)
5 \( 1 + iT \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (-0.233 - 0.972i)T \)
19 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + (-0.233 + 0.972i)T \)
29 \( 1 + (0.891 + 0.453i)T \)
31 \( 1 + (-0.891 + 0.453i)T \)
37 \( 1 + (-0.760 + 0.649i)T \)
41 \( 1 + (0.0784 - 0.996i)T \)
43 \( 1 + (0.972 + 0.233i)T \)
47 \( 1 + (-0.453 - 0.891i)T \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (-0.453 - 0.891i)T \)
67 \( 1 + (-0.852 + 0.522i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.891 - 0.453i)T \)
89 \( 1 + (-0.453 - 0.891i)T \)
97 \( 1 + (0.987 - 0.156i)T \)
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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

−17.9052633340553838448553372242, −17.43212150555565078054940343058, −16.682844897787119273667482007792, −16.146501454765310841106262351163, −15.348804506766967892378171689372, −14.88564793939811705316745008424, −14.135244030369047900785379716453, −13.20208885456002424923123784625, −12.5143431080587506954115777392, −12.18390513856823043476550407190, −11.25875404698092040088302193906, −10.91545353946239794010846445829, −10.17422254269110917740109276371, −9.223830671742548605475114551465, −8.49287386514609089750045183061, −7.67539387856885897591935726429, −6.49874364775330788937671450373, −5.882903164046553735122179743278, −5.36072955116149260480724184163, −4.497060667008741157851279310335, −4.337761632694678754903689514977, −3.24241324852621649651768129910, −2.1848175354269928337522513847, −1.28706346804341657774439417894, −0.434418207461706462745515912211, 1.67896006277884755246263823623, 1.88785718402307056410254329244, 2.962708654690473504188361180205, 4.07581480066592339444328464900, 4.61732804303841089641391023123, 5.31432931121540024240228823226, 6.144502171341823607174313205898, 6.88443297821207865793054978420, 7.33972639255054261212223463100, 7.72188559600666377219658673270, 8.983920877028574227658643455486, 10.26045126924120329477353273218, 10.642423016425116736354234874685, 11.60213519768614675765953722680, 11.83171427884679748893945038890, 12.46283957398059794542914824020, 13.48727028257319545658071217, 14.13013925016687123466231094399, 14.45998378716246088106319630763, 15.27885721171011499828274164812, 15.997411726530468441773155199917, 16.80244453051939343537731654488, 17.48861440596444584466161780295, 17.80957365234731232642903713685, 18.669149934519115111656737698476

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