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Properties

Label 1-6001-6001.49-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.869 - 0.493i$
Analytic cond. $27.8685$
Root an. cond. $27.8685$
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  − 2-s + (0.923 − 0.382i)3-s + 4-s + (−0.382 − 0.923i)5-s + (−0.923 + 0.382i)6-s + (0.382 − 0.923i)7-s − 8-s + (0.707 − 0.707i)9-s + (0.382 + 0.923i)10-s + (−0.707 + 0.707i)11-s + (0.923 − 0.382i)12-s + (0.923 + 0.382i)13-s + (−0.382 + 0.923i)14-s + (−0.707 − 0.707i)15-s + 16-s + ⋯
L(s)  = 1  − 2-s + (0.923 − 0.382i)3-s + 4-s + (−0.382 − 0.923i)5-s + (−0.923 + 0.382i)6-s + (0.382 − 0.923i)7-s − 8-s + (0.707 − 0.707i)9-s + (0.382 + 0.923i)10-s + (−0.707 + 0.707i)11-s + (0.923 − 0.382i)12-s + (0.923 + 0.382i)13-s + (−0.382 + 0.923i)14-s + (−0.707 − 0.707i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.869 - 0.493i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.869 - 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3614223758 - 1.369552353i\)
\(L(\frac12)\) \(\approx\) \(0.3614223758 - 1.369552353i\)
\(L(1)\) \(\approx\) \(0.8186573909 - 0.4576232368i\)
\(L(1)\) \(\approx\) \(0.8186573909 - 0.4576232368i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (0.382 - 0.923i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.923 + 0.382i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.923 - 0.382i)T \)
59 \( 1 + (0.923 - 0.382i)T \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 + (-0.923 - 0.382i)T \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.923 - 0.382i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−18.25696735072603941755425282808, −17.56609662483273920334355061232, −16.44218583915503523502808829614, −15.99618001332502003765962720394, −15.37713182499664219032444264870, −14.91871489651437551459486243578, −14.394146026789839550216093593551, −13.48382626086087676233531659857, −12.659134920056118771704676323370, −11.885576174396360882024499591510, −10.86164207695875545866708238338, −10.79512900947053969325733421697, −10.12847622367221998533846721999, −9.0080347889334979191170233908, −8.70929193743292927175199067962, −8.16033687128429942286986696148, −7.4547075376524258641619745468, −6.833730553213690300808923267637, −5.82510704970405569796742381138, −5.319655280968330743359159636514, −3.906835342218172492975655051753, −3.328653983985001786821059689555, −2.66867578257460576869307927067, −2.11507967376533039794878151971, −1.14783186633604495345795557096, 0.467555343955090356106812904558, 1.16155789400909503386432840462, 1.93338093028920350755843460533, 2.57463724526651541709541900618, 3.7415127237196885003873535986, 4.174037510439015029769266780039, 5.14916778314651443799374699697, 6.210743762818182428736973451506, 7.10049683243317005786094730953, 7.51009099141044104518352999343, 8.10757764580016479810943528254, 8.71518654113478521917959785302, 9.305975817602869844809320074480, 9.86982599018985487959298693691, 10.909289033185774852453199379794, 11.20470384030070397088747202557, 12.24035204126667174169975279830, 12.969123137385323438336179021201, 13.26933880136642652171179785959, 14.23137119742022689474741847318, 15.06229669911033978121951288052, 15.51274621976607285970605121932, 16.17259542189026199151322348056, 16.90704364243664094310088827275, 17.50783678091912431826299861504

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