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Properties

Label 1-6001-6001.56-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.196 - 0.980i$
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  + (−0.142 − 0.989i)2-s + (0.998 − 0.0535i)3-s + (−0.959 + 0.281i)4-s + (−0.668 + 0.743i)5-s + (−0.195 − 0.980i)6-s + (−0.555 − 0.831i)7-s + (0.415 + 0.909i)8-s + (0.994 − 0.106i)9-s + (0.831 + 0.555i)10-s + (0.894 − 0.447i)11-s + (−0.943 + 0.332i)12-s + (0.916 + 0.399i)13-s + (−0.743 + 0.668i)14-s + (−0.627 + 0.778i)15-s + (0.841 − 0.540i)16-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (0.998 − 0.0535i)3-s + (−0.959 + 0.281i)4-s + (−0.668 + 0.743i)5-s + (−0.195 − 0.980i)6-s + (−0.555 − 0.831i)7-s + (0.415 + 0.909i)8-s + (0.994 − 0.106i)9-s + (0.831 + 0.555i)10-s + (0.894 − 0.447i)11-s + (−0.943 + 0.332i)12-s + (0.916 + 0.399i)13-s + (−0.743 + 0.668i)14-s + (−0.627 + 0.778i)15-s + (0.841 − 0.540i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.196 - 0.980i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.196 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.481277850 - 1.807457884i\)
\(L(\frac12)\) \(\approx\) \(1.481277850 - 1.807457884i\)
\(L(1)\) \(\approx\) \(1.132331200 - 0.6545306703i\)
\(L(1)\) \(\approx\) \(1.132331200 - 0.6545306703i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.142 - 0.989i)T \)
3 \( 1 + (0.998 - 0.0535i)T \)
5 \( 1 + (-0.668 + 0.743i)T \)
7 \( 1 + (-0.555 - 0.831i)T \)
11 \( 1 + (0.894 - 0.447i)T \)
13 \( 1 + (0.916 + 0.399i)T \)
19 \( 1 + (0.821 - 0.570i)T \)
23 \( 1 + (0.0713 - 0.997i)T \)
29 \( 1 + (0.994 - 0.106i)T \)
31 \( 1 + (0.964 - 0.264i)T \)
37 \( 1 + (0.973 - 0.229i)T \)
41 \( 1 + (-0.997 + 0.0713i)T \)
43 \( 1 + (-0.894 + 0.447i)T \)
47 \( 1 + (-0.106 + 0.994i)T \)
53 \( 1 + (0.668 - 0.743i)T \)
59 \( 1 + (0.831 + 0.555i)T \)
61 \( 1 + (-0.177 - 0.984i)T \)
67 \( 1 + (0.831 - 0.555i)T \)
71 \( 1 + (0.641 - 0.767i)T \)
73 \( 1 + (0.627 + 0.778i)T \)
79 \( 1 + (-0.668 - 0.743i)T \)
83 \( 1 + (-0.909 - 0.415i)T \)
89 \( 1 + (-0.159 - 0.987i)T \)
97 \( 1 + (-0.731 + 0.681i)T \)
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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

−18.00753155854080018839948286249, −17.05203252113740275904771170880, −16.352301499997582716437491399129, −15.83778374020830312782985184455, −15.272658685568200342046364481506, −14.99920775099826306037510890830, −13.91977284665984947944733213683, −13.55545165754292562425979255270, −12.784107351191599080070536935531, −12.187662699778713121572072849, −11.50830364977405555305337345054, −10.0638492594302695836211726397, −9.73632715823632051286636625713, −8.99420413244351099253324978185, −8.38800929453873643822913586178, −8.16932854923009536151473255163, −7.13397450309518799769600124194, −6.658419582371172626427107156859, −5.66638654284981951097960097150, −5.07170629475630558023410725376, −4.11473343368828321503920321854, −3.66847297091439695167364125188, −2.91969546939294162947909705972, −1.549217482678919295765400741111, −0.996032268297627307240328026340, 0.713269399510760745556143632641, 1.28023227112232450487938579733, 2.45687456517079975531684437132, 3.056513621708026435036687485111, 3.62603997489411812313653566669, 4.12050345377578153952900145994, 4.777640344998991702449977658345, 6.43939744584843146177354589029, 6.691492266766235983778062366348, 7.71448622901815268377395222111, 8.27503138189769626123030868713, 8.89530010040886571598910662526, 9.68419021795519593010992010845, 10.17644154809196984524171445667, 10.925662555537279942931585206347, 11.52778525826369768764369311939, 12.14455073476969698899260270309, 13.10337212535369232329766890817, 13.548741409546741034076760111829, 14.20875743280595725626133210425, 14.52623030154112360549267398314, 15.578706506679699412138447479037, 16.175349780477658205091572167673, 16.8942963915114277236853901276, 17.83770124675499233748110965687

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