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Properties

Label 1-6001-6001.108-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.566 - 0.823i$
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  + (−0.540 + 0.841i)2-s + (0.719 − 0.694i)3-s + (−0.415 − 0.909i)4-s + (−0.0178 + 0.999i)5-s + (0.195 + 0.980i)6-s + (−0.831 + 0.555i)7-s + (0.989 + 0.142i)8-s + (0.0356 − 0.999i)9-s + (−0.831 − 0.555i)10-s + (0.994 − 0.106i)11-s + (−0.930 − 0.366i)12-s + (−0.613 − 0.789i)13-s + (−0.0178 − 0.999i)14-s + (0.681 + 0.731i)15-s + (−0.654 + 0.755i)16-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)2-s + (0.719 − 0.694i)3-s + (−0.415 − 0.909i)4-s + (−0.0178 + 0.999i)5-s + (0.195 + 0.980i)6-s + (−0.831 + 0.555i)7-s + (0.989 + 0.142i)8-s + (0.0356 − 0.999i)9-s + (−0.831 − 0.555i)10-s + (0.994 − 0.106i)11-s + (−0.930 − 0.366i)12-s + (−0.613 − 0.789i)13-s + (−0.0178 − 0.999i)14-s + (0.681 + 0.731i)15-s + (−0.654 + 0.755i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.566 - 0.823i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.566 - 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2298859622 - 0.4372898167i\)
\(L(\frac12)\) \(\approx\) \(0.2298859622 - 0.4372898167i\)
\(L(1)\) \(\approx\) \(0.8103587153 + 0.1007258736i\)
\(L(1)\) \(\approx\) \(0.8103587153 + 0.1007258736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.540 + 0.841i)T \)
3 \( 1 + (0.719 - 0.694i)T \)
5 \( 1 + (-0.0178 + 0.999i)T \)
7 \( 1 + (-0.831 + 0.555i)T \)
11 \( 1 + (0.994 - 0.106i)T \)
13 \( 1 + (-0.613 - 0.789i)T \)
19 \( 1 + (0.948 - 0.315i)T \)
23 \( 1 + (-0.281 - 0.959i)T \)
29 \( 1 + (0.731 - 0.681i)T \)
31 \( 1 + (0.767 - 0.641i)T \)
37 \( 1 + (-0.943 - 0.332i)T \)
41 \( 1 + (-0.281 + 0.959i)T \)
43 \( 1 + (-0.778 - 0.627i)T \)
47 \( 1 + (-0.0356 - 0.999i)T \)
53 \( 1 + (0.694 + 0.719i)T \)
59 \( 1 + (0.195 + 0.980i)T \)
61 \( 1 + (-0.948 - 0.315i)T \)
67 \( 1 + (-0.980 - 0.195i)T \)
71 \( 1 + (-0.850 - 0.525i)T \)
73 \( 1 + (-0.999 + 0.0356i)T \)
79 \( 1 + (0.0178 + 0.999i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
89 \( 1 + (0.0535 - 0.998i)T \)
97 \( 1 + (-0.860 + 0.510i)T \)
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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

−17.78173539246409596088787327247, −17.28441334639806816617615013793, −16.493603252793102504778564703277, −16.28614812942995296529558436315, −15.59343773815251770073140768415, −14.45819316573425410093623646518, −13.71051748296533260113418682246, −13.56776931740868719498586014597, −12.45979567518356546616226430120, −12.05121251046380260711096052291, −11.37506273277293263531935164591, −10.33853209938887792316759003068, −9.84272030356351932061639611680, −9.3865844485910042992676393901, −8.87628582419283919539091783189, −8.21184958571625327675630468632, −7.37732096791220183183441456856, −6.772267360572081247185049337065, −5.445221128621066029702844034821, −4.61920961170451758380827085969, −4.13255435138510619579242678318, −3.431688076882436023749533454495, −2.87388536515638541171565691286, −1.641122695791691992530770466754, −1.31139704521194062677945601233, 0.140234977161812743699838527651, 1.14569437109812201319677521595, 2.21940539907424968056134162262, 2.838446226460162511688981145113, 3.55019552062407348047187474468, 4.5219456787331350519098336500, 5.73821554249352933376714844046, 6.18996336023620845340083062789, 6.86342271133314102096886057847, 7.259042564733369371165952047102, 8.07659344803152780474104860375, 8.69535617487518601385705421390, 9.40674224018255459331958760960, 9.98543009853141697762344207199, 10.50313258233308923766849237426, 11.90596613807567421385823999583, 11.973672912033770437357313340429, 13.27779232439182846240554070102, 13.64830173438698383554012637257, 14.361855557397896377562774891340, 15.02032260580879378464144981639, 15.29341197689625869577871059984, 16.077454567896902056505377698391, 16.9542587559075429661264420206, 17.61789615147817711513219105160

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