Properties

Label 1-6001-6001.4583-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.693 - 0.720i$
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 + (0.555 + 0.831i)3-s + 4-s + (−0.195 + 0.980i)5-s + (−0.555 − 0.831i)6-s + (0.980 − 0.195i)7-s − 8-s + (−0.382 + 0.923i)9-s + (0.195 − 0.980i)10-s + (−0.923 + 0.382i)11-s + (0.555 + 0.831i)12-s + (0.195 + 0.980i)13-s + (−0.980 + 0.195i)14-s + (−0.923 + 0.382i)15-s + 16-s + ⋯
L(s)  = 1  − 2-s + (0.555 + 0.831i)3-s + 4-s + (−0.195 + 0.980i)5-s + (−0.555 − 0.831i)6-s + (0.980 − 0.195i)7-s − 8-s + (−0.382 + 0.923i)9-s + (0.195 − 0.980i)10-s + (−0.923 + 0.382i)11-s + (0.555 + 0.831i)12-s + (0.195 + 0.980i)13-s + (−0.980 + 0.195i)14-s + (−0.923 + 0.382i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.693 - 0.720i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (4583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.693 - 0.720i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2794261191 - 0.1189714122i\)
\(L(\frac12)\) \(\approx\) \(0.2794261191 - 0.1189714122i\)
\(L(1)\) \(\approx\) \(0.6170675034 + 0.3179696351i\)
\(L(1)\) \(\approx\) \(0.6170675034 + 0.3179696351i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.555 + 0.831i)T \)
5 \( 1 + (-0.195 + 0.980i)T \)
7 \( 1 + (0.980 - 0.195i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + (0.195 + 0.980i)T \)
19 \( 1 + (-0.923 + 0.382i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (-0.923 - 0.382i)T \)
31 \( 1 + (0.195 - 0.980i)T \)
37 \( 1 + (0.555 - 0.831i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 + (0.923 - 0.382i)T \)
53 \( 1 + (-0.980 - 0.195i)T \)
59 \( 1 + (0.980 + 0.195i)T \)
61 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 + (-0.980 + 0.195i)T \)
71 \( 1 + (-0.195 - 0.980i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.195 - 0.980i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.980 + 0.195i)T \)
97 \( 1 + (0.923 - 0.382i)T \)
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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

−17.83077371760260672942239309477, −17.35081902612045229447946598151, −16.73609031675286903349579660098, −15.66929923453970418869867810425, −15.45701467377857001853371337007, −14.66760568248045631552413482537, −13.79030213944337560981831124478, −13.024536438757064250709095760685, −12.56180629784551326942988860366, −11.82429153364323124201966582341, −11.20727394640178787454682993711, −10.50148459328394955119327659732, −9.641604718099684738576676334831, −8.75963092904816782962803361342, −8.47552811214948331586233460993, −7.84372308683647246269898921278, −7.53182146773752472244558883262, −6.46657001723795669522549802184, −5.6374645928239014026101354425, −5.137344948015538896575765840475, −3.92085421315310858266606508317, −3.01359105826481325020438474484, −2.26718453042071323449001011137, −1.53691707348794263484179362183, −0.92413497840773886204778702748, 0.102210993526511567144455716707, 1.84394714598775979472550046062, 2.179206118468131815450789774, 2.87055296154075817348088276121, 4.013910349746055832263658698353, 4.309767861008180621908025770485, 5.548834336167621468039707755361, 6.203355734499439136969936406426, 7.2552757086324603593870211331, 7.664733068212994856950125312248, 8.30021656151429553512471185123, 8.90779775060558208036987520387, 9.78921220109878920857252983726, 10.30228078879360832893670129566, 10.846027334970188000677452132511, 11.33128441398602992717972422728, 11.98865275176204902380699538435, 13.126565162555522518409210106482, 14.02503445583808862765408907051, 14.69915678443929837992018224742, 14.96054490708875335412652862146, 15.68743988044625530736315781828, 16.34488261263010229240999922511, 16.95248098585775931812850092026, 17.71915301494626227368918768771

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