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Properties

Label 1-6001-6001.60-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.835 - 0.549i$
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 + (−0.382 + 0.923i)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 + (−0.382 + 0.923i)13-s + (0.382 − 0.923i)14-s + (−0.707 − 0.707i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.835 - 0.549i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.835 - 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.271507546 - 4.248670985i\)
\(L(\frac12)\) \(\approx\) \(1.271507546 - 4.248670985i\)
\(L(1)\) \(\approx\) \(2.061899559 - 1.283922647i\)
\(L(1)\) \(\approx\) \(2.061899559 - 1.283922647i\)

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 \( 1 + (-0.382 + 0.923i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.382 + 0.923i)T \)
59 \( 1 + (0.382 + 0.923i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + (0.382 - 0.923i)T \)
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 \( 1 + (0.382 - 0.923i)T \)
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

−18.069257876426828033364695778249, −17.3822946369579683552126045215, −16.166066623512195659498808465995, −15.693930038505749880034684403129, −15.17578120800811840462121158582, −14.64366945956658579202197031424, −14.45491506943909668211012824756, −13.36700127159776470754390903745, −12.82822057385985888347048716019, −12.2527498739527599896166678229, −11.34481720028214595248596244211, −10.82790112337665893416703412990, −10.05958035919766583639458288289, −9.57339446017196839264683133773, −8.28568144291847916132029126485, −7.86516516205354988219180427823, −7.3611714118186747197888132897, −6.46928755834780773255511045097, −5.55013657940389579022209784607, −5.062297807041964602459337757021, −4.15505542495348234713549776520, −3.53929437852995719396370614568, −2.782839887463262322711671187383, −2.3309107220859405329753107230, −1.67493546703497340792107526791, 0.57104498965217739316298156067, 1.45636484517106459574721049042, 2.245797091021060464778218935995, 2.92947974781947758534590141726, 4.07335681330159777131925676837, 4.13861598823709885617767291878, 4.96656606776827796298249085509, 5.83749545546086511999989069769, 6.81515312549937540407703457395, 7.29722128408339866350217431639, 7.99529292952744948612682369002, 8.55586559825930138974773933707, 9.33729567571672259469175643246, 10.26056447882731805456727265411, 11.06487798254563648042202549324, 11.59318139047007735628664758555, 12.52400483944913308685220616706, 12.97283802933310716854694187415, 13.48883298723368766136166921665, 14.04858023098752464111857637897, 14.7467850355390547993805498473, 15.2399308221175340072654950808, 16.14664817708757050611189371787, 16.59544668703973504846614147146, 17.14012681549607665149664584684

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