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Properties

Label 1-6001-6001.363-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.968 + 0.250i$
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 + (−0.195 + 0.980i)3-s + 4-s + (0.831 − 0.555i)5-s + (0.195 − 0.980i)6-s + (−0.555 + 0.831i)7-s − 8-s + (−0.923 − 0.382i)9-s + (−0.831 + 0.555i)10-s + (0.382 + 0.923i)11-s + (−0.195 + 0.980i)12-s + (−0.831 − 0.555i)13-s + (0.555 − 0.831i)14-s + (0.382 + 0.923i)15-s + 16-s + ⋯
L(s)  = 1  − 2-s + (−0.195 + 0.980i)3-s + 4-s + (0.831 − 0.555i)5-s + (0.195 − 0.980i)6-s + (−0.555 + 0.831i)7-s − 8-s + (−0.923 − 0.382i)9-s + (−0.831 + 0.555i)10-s + (0.382 + 0.923i)11-s + (−0.195 + 0.980i)12-s + (−0.831 − 0.555i)13-s + (0.555 − 0.831i)14-s + (0.382 + 0.923i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.968 + 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9591715576 + 0.1221269750i\)
\(L(\frac12)\) \(\approx\) \(0.9591715576 + 0.1221269750i\)
\(L(1)\) \(\approx\) \(0.6725132326 + 0.1922759483i\)
\(L(1)\) \(\approx\) \(0.6725132326 + 0.1922759483i\)

Euler product

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

−17.53022087197211913255002746467, −17.34728039938643416744212436090, −16.45690511093002126927397107388, −16.29553197915339506343120478097, −14.920999584376970883124543961902, −14.375706530946627084071799387571, −13.75362329868890857304985252326, −13.07850995484065677372155692546, −12.40773149060383582534096849479, −11.54206517083912293938865514507, −10.92772783793094593338618837635, −10.55114339681518164570937027835, −9.51375467030107101544861439371, −9.17205822557956977481460140920, −8.329411270107805438918436502255, −7.382563647525415157205901977910, −6.95863911176455296716314453592, −6.541160217427016578303720676860, −5.854182817670728716501576526741, −5.013995277835295903578006111259, −3.61251285613228463258730024930, −2.75393018622278834102869594602, −2.3943759647578432333599349416, −1.26857913827211462584539534805, −0.78691477329900738638619712633, 0.45918586145186792659068969747, 1.644980857294901717537118830193, 2.36525354334747532904002021270, 3.05706054922075601427099990114, 3.97339553451827353253342089655, 5.08534521455170184227850238084, 5.54288615461492417877502360513, 6.144175545200263643267631819224, 6.99130271745639994432450689325, 7.86636791627286781129146289930, 8.69968886333594788515328235594, 9.34460833831736528166384657005, 9.65384940843609839329714704085, 10.119341742705942452748828065825, 10.884157857801966821553480088555, 11.78324027015005187175047775510, 12.37688257489457757433212581096, 12.77065465604383962683435583770, 14.0631509202856782424310257387, 14.77035144977835690919963487328, 15.285068817910090757203369826343, 15.95115532021950249725296183855, 16.4986494325211231554756608406, 17.16400105548543060376360731569, 17.607475066989380510890168328007

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