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Properties

Label 1-6001-6001.1063-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.945 - 0.325i$
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  + (−0.909 − 0.415i)2-s + (0.936 + 0.349i)3-s + (0.654 + 0.755i)4-s + (−0.936 − 0.349i)5-s + (−0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + (−0.281 − 0.959i)8-s + (0.755 + 0.654i)9-s + (0.707 + 0.707i)10-s + (0.212 − 0.977i)11-s + (0.349 + 0.936i)12-s + (−0.989 − 0.142i)13-s + (0.349 + 0.936i)14-s + (−0.755 − 0.654i)15-s + (−0.142 + 0.989i)16-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)2-s + (0.936 + 0.349i)3-s + (0.654 + 0.755i)4-s + (−0.936 − 0.349i)5-s + (−0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + (−0.281 − 0.959i)8-s + (0.755 + 0.654i)9-s + (0.707 + 0.707i)10-s + (0.212 − 0.977i)11-s + (0.349 + 0.936i)12-s + (−0.989 − 0.142i)13-s + (0.349 + 0.936i)14-s + (−0.755 − 0.654i)15-s + (−0.142 + 0.989i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.945 - 0.325i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (1063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.945 - 0.325i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1242118906 - 0.7414795517i\)
\(L(\frac12)\) \(\approx\) \(0.1242118906 - 0.7414795517i\)
\(L(1)\) \(\approx\) \(0.6796711358 - 0.2511058407i\)
\(L(1)\) \(\approx\) \(0.6796711358 - 0.2511058407i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.909 - 0.415i)T \)
3 \( 1 + (0.936 + 0.349i)T \)
5 \( 1 + (-0.936 - 0.349i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.212 - 0.977i)T \)
13 \( 1 + (-0.989 - 0.142i)T \)
19 \( 1 + (0.989 - 0.142i)T \)
23 \( 1 + (-0.977 + 0.212i)T \)
29 \( 1 + (0.997 - 0.0713i)T \)
31 \( 1 + (0.212 - 0.977i)T \)
37 \( 1 + (0.479 - 0.877i)T \)
41 \( 1 + (0.212 - 0.977i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 + (-0.654 - 0.755i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.599 - 0.800i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.997 + 0.0713i)T \)
73 \( 1 + (-0.0713 + 0.997i)T \)
79 \( 1 + (0.936 - 0.349i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
89 \( 1 + (0.281 - 0.959i)T \)
97 \( 1 + (0.479 + 0.877i)T \)
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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

−18.151360832981043192752972784487, −17.63890891823340567964576192615, −16.46034303750727379406566947789, −16.00938895564037003603870177175, −15.438563866327592096190050164973, −14.734728379465042204998879951920, −14.53946367083119364610080676815, −13.57390941757327276231515255763, −12.47245402376337838877920022980, −12.09575840625964106413380834977, −11.58464103775394232725236606275, −10.32781729335993797398020438667, −9.820037724708397962061105796511, −9.39065912813934223792031187629, −8.54394252023879603749631108816, −7.91920448072194301911197669331, −7.48031354589391023168316185584, −6.635146195336840814817806454613, −6.41152880725489954605639678186, −5.04811480597915001849594542972, −4.37560948990668410423813358379, −3.15456092245364608849441251449, −2.8059773543872139857678148238, −1.96389376310856698316655748380, −1.06719841979241677222818407170, 0.27820755728567052553625227660, 0.988840875978609095086349659410, 2.13962729434348807684221625980, 2.94847912081552150376986782862, 3.55652489059476647068428641517, 3.9930038801599863748674430743, 4.88875497564517887308475210587, 6.10250528507241540448386733904, 7.120849890055751299858917611405, 7.53477039716655678041784838593, 8.11433735735713789980522087708, 8.76039958069159550491567038532, 9.503764681873680091094874151714, 9.90823984548977970303221544636, 10.68411091597393216801358648477, 11.320204626519035560768138051026, 12.19029405280702780476228051178, 12.58336879111069358198393385455, 13.643478033540964441219181255965, 13.93083226805740382779359124084, 15.07587248826541118873697157578, 15.65832071658689565970901645493, 16.17184916509344398044882887736, 16.67770028020333496228600073742, 17.27116425013559712478705191249

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