Properties

Label 1-6017-6017.197-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.908 - 0.418i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
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  + (0.568 − 0.822i)2-s − 3-s + (−0.354 − 0.935i)4-s + (0.970 − 0.239i)5-s + (−0.568 + 0.822i)6-s + (0.568 + 0.822i)7-s + (−0.970 − 0.239i)8-s + 9-s + (0.354 − 0.935i)10-s + (0.354 + 0.935i)12-s − 13-s + 14-s + (−0.970 + 0.239i)15-s + (−0.748 + 0.663i)16-s + (0.120 − 0.992i)17-s + (0.568 − 0.822i)18-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)2-s − 3-s + (−0.354 − 0.935i)4-s + (0.970 − 0.239i)5-s + (−0.568 + 0.822i)6-s + (0.568 + 0.822i)7-s + (−0.970 − 0.239i)8-s + 9-s + (0.354 − 0.935i)10-s + (0.354 + 0.935i)12-s − 13-s + 14-s + (−0.970 + 0.239i)15-s + (−0.748 + 0.663i)16-s + (0.120 − 0.992i)17-s + (0.568 − 0.822i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.908 - 0.418i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.908 - 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3345475267 - 1.526062706i\)
\(L(\frac12)\) \(\approx\) \(0.3345475267 - 1.526062706i\)
\(L(1)\) \(\approx\) \(0.9486572391 - 0.6259817353i\)
\(L(1)\) \(\approx\) \(0.9486572391 - 0.6259817353i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.568 - 0.822i)T \)
3 \( 1 - T \)
5 \( 1 + (0.970 - 0.239i)T \)
7 \( 1 + (0.568 + 0.822i)T \)
13 \( 1 - T \)
17 \( 1 + (0.120 - 0.992i)T \)
19 \( 1 + (0.748 + 0.663i)T \)
23 \( 1 + (-0.885 + 0.464i)T \)
29 \( 1 + (-0.885 - 0.464i)T \)
31 \( 1 + (0.748 - 0.663i)T \)
37 \( 1 + (-0.885 - 0.464i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.354 - 0.935i)T \)
47 \( 1 + (0.885 + 0.464i)T \)
53 \( 1 + (-0.970 + 0.239i)T \)
59 \( 1 + (0.748 - 0.663i)T \)
61 \( 1 + (0.568 - 0.822i)T \)
67 \( 1 + (-0.354 - 0.935i)T \)
71 \( 1 + (0.748 + 0.663i)T \)
73 \( 1 + (0.970 + 0.239i)T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
89 \( 1 + (-0.568 - 0.822i)T \)
97 \( 1 + (0.120 - 0.992i)T \)
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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

−17.60973339441822009889481727614, −17.43967833015374534072133762934, −16.65965871217571291959891222053, −16.29359175581792037719409485171, −15.29679216568400614316732382396, −14.713227801958183954848638440459, −14.04900357328402883013240700771, −13.53207013727114994852833295337, −12.775471465624011831815056061110, −12.26553107979061710398194184849, −11.45216067419505435695827586239, −10.742877134468653190984694044, −10.055955458455076182305707455615, −9.4970434476770640568780694200, −8.44505142547184817187440955367, −7.65644709408833889173777042132, −6.95256234343857149353838877302, −6.60862555127948509498133013749, −5.651981321484559854230399698221, −5.285789843855115857137382147274, −4.54517510778249646289846915461, −3.94225693295093357600981268084, −2.88269126783236049045008111682, −1.91583766538586159200690044573, −0.998184218629440741246631909257, 0.391808961303858968249731268832, 1.41604477862072285582362675435, 2.06177799536121872862487792987, 2.62936268813371883765006378998, 3.78404397877366168525639199420, 4.65847108947368530589243924716, 5.26375559819541696335096527981, 5.60613327564099778750725359806, 6.20641220168385571042242553920, 7.165039476675433450416734492738, 8.0385306769738216347706523126, 9.28912701642091929705133223615, 9.57631607999920753574439292822, 10.14504800040666544989063145097, 10.98177072225655897766980119847, 11.64031782732155414697227085405, 12.14275250283630899130808107180, 12.569157518004540234414928531962, 13.38974235850837828847333099080, 14.12518978807457318746259218056, 14.507519519289177284166383700855, 15.60780836615880804930264801350, 15.90477413883440474652824898040, 17.163222151659883131023182393610, 17.35000756209379462790180303302

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