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Properties

Label 1-6001-6001.331-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.147 - 0.989i$
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  + (0.755 + 0.654i)2-s + (0.0713 − 0.997i)3-s + (0.142 + 0.989i)4-s + (−0.0713 + 0.997i)5-s + (0.707 − 0.707i)6-s + (−0.707 + 0.707i)7-s + (−0.540 + 0.841i)8-s + (−0.989 − 0.142i)9-s + (−0.707 + 0.707i)10-s + (−0.936 − 0.349i)11-s + (0.997 − 0.0713i)12-s + (−0.959 + 0.281i)13-s + (−0.997 + 0.0713i)14-s + (0.989 + 0.142i)15-s + (−0.959 + 0.281i)16-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)2-s + (0.0713 − 0.997i)3-s + (0.142 + 0.989i)4-s + (−0.0713 + 0.997i)5-s + (0.707 − 0.707i)6-s + (−0.707 + 0.707i)7-s + (−0.540 + 0.841i)8-s + (−0.989 − 0.142i)9-s + (−0.707 + 0.707i)10-s + (−0.936 − 0.349i)11-s + (0.997 − 0.0713i)12-s + (−0.959 + 0.281i)13-s + (−0.997 + 0.0713i)14-s + (0.989 + 0.142i)15-s + (−0.959 + 0.281i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.147 - 0.989i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.147 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3888335217 + 0.3350726398i\)
\(L(\frac12)\) \(\approx\) \(-0.3888335217 + 0.3350726398i\)
\(L(1)\) \(\approx\) \(0.8230153819 + 0.5776849313i\)
\(L(1)\) \(\approx\) \(0.8230153819 + 0.5776849313i\)

Euler product

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

−16.90608157180164799387704141260, −16.55606479605898149440298753096, −15.70221725575440979338082031061, −15.27622356104659184423369323499, −14.6721199845089349324203682384, −13.75035745918484147694309747093, −13.17545576124329627377013921202, −12.72924338991116543236301245071, −12.02110090250468695522079471177, −11.183631285854585059109248007692, −10.54065041844638128827195057877, −10.02692516383622027155407450231, −9.39573925162359421046386944739, −8.89825934136727130312451496442, −7.792430364666790037217224082867, −7.0736096545987668486316300367, −5.97700555097303821604697324059, −5.36665585392109646624863019883, −4.74957457591781093374023446466, −4.223002216947975375417512052839, −3.6289339470029803714597724216, −2.59222231603454217243306554940, −2.23772927827244743999489573972, −0.66881528019243257976790179996, −0.127449962905772390466095936490, 1.76824517410743701741852394799, 2.46440556544066476773568687787, 3.14583781661355191474904062822, 3.48192767941200101160456144101, 4.85245437286891187172162906067, 5.582619473722151625709543043648, 6.123747197343242526530838939277, 6.710089623059379078432454155899, 7.4401514255996822399467372397, 7.79161293830724231378291494961, 8.63791833449659547992330780602, 9.43708673360861659478140588241, 10.35849842704093146010126859073, 11.31051086223221954499073142191, 11.768099322976306448229871387805, 12.61936008353204883845920370004, 12.882723957554145536189903783840, 13.71898612815619346154740964260, 14.2536794194050762280878648545, 14.97627682239124765111411355674, 15.29552634202584629265711127348, 16.31341094517799533211870611156, 16.71773178810547639825278802049, 17.73018522254568949313687568218, 18.14770733543830215255404720592

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