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Properties

Label 1-6017-6017.348-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.862 - 0.505i$
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  + (0.899 + 0.435i)2-s + (0.809 + 0.587i)3-s + (0.619 + 0.784i)4-s + (−0.594 + 0.804i)5-s + (0.471 + 0.881i)6-s + (0.339 + 0.940i)7-s + (0.215 + 0.976i)8-s + (0.309 + 0.951i)9-s + (−0.885 + 0.464i)10-s + (0.0402 + 0.999i)12-s + (0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.953 + 0.301i)15-s + (−0.231 + 0.972i)16-s + (−0.932 + 0.362i)17-s + (−0.136 + 0.990i)18-s + ⋯
L(s)  = 1  + (0.899 + 0.435i)2-s + (0.809 + 0.587i)3-s + (0.619 + 0.784i)4-s + (−0.594 + 0.804i)5-s + (0.471 + 0.881i)6-s + (0.339 + 0.940i)7-s + (0.215 + 0.976i)8-s + (0.309 + 0.951i)9-s + (−0.885 + 0.464i)10-s + (0.0402 + 0.999i)12-s + (0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.953 + 0.301i)15-s + (−0.231 + 0.972i)16-s + (−0.932 + 0.362i)17-s + (−0.136 + 0.990i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.862 - 0.505i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.862 - 0.505i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9955625350 + 3.669346411i\)
\(L(\frac12)\) \(\approx\) \(-0.9955625350 + 3.669346411i\)
\(L(1)\) \(\approx\) \(1.290074782 + 1.712317124i\)
\(L(1)\) \(\approx\) \(1.290074782 + 1.712317124i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.899 + 0.435i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.594 + 0.804i)T \)
7 \( 1 + (0.339 + 0.940i)T \)
13 \( 1 + (0.978 + 0.207i)T \)
17 \( 1 + (-0.932 + 0.362i)T \)
19 \( 1 + (-0.759 - 0.650i)T \)
23 \( 1 + (0.919 - 0.391i)T \)
29 \( 1 + (0.681 + 0.732i)T \)
31 \( 1 + (0.607 - 0.794i)T \)
37 \( 1 + (-0.974 + 0.223i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.845 + 0.534i)T \)
47 \( 1 + (-0.999 + 0.0161i)T \)
53 \( 1 + (0.870 - 0.493i)T \)
59 \( 1 + (-0.759 + 0.650i)T \)
61 \( 1 + (-0.471 - 0.881i)T \)
67 \( 1 + (-0.845 + 0.534i)T \)
71 \( 1 + (-0.991 + 0.128i)T \)
73 \( 1 + (0.581 - 0.813i)T \)
79 \( 1 + (-0.168 + 0.985i)T \)
83 \( 1 + (-0.993 - 0.112i)T \)
89 \( 1 + (0.970 - 0.239i)T \)
97 \( 1 + (0.457 + 0.889i)T \)
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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

−17.3540385807598042400087120773, −16.51310518012820104598241709495, −15.72649978069885456481412222978, −15.32235374378975273368952840188, −14.5426399628290436134281266836, −13.83887124630157244580848780516, −13.31529282341664959435194232201, −12.99573311656515472829660519972, −12.14194687621309617210462534094, −11.61110719854856193662438350460, −10.83139129432411346777849991902, −10.21799356192548625284676979963, −9.2197199586620252303382887323, −8.570152873055112722951988426627, −7.94531745711945914718400871790, −7.14412559319643262550556471778, −6.60025850998098368701168664202, −5.757133314605397295500064303127, −4.65701368074612298996065596859, −4.32647580301088331166716975905, −3.53802443738592302886191205855, −2.99601764050996846940306847426, −1.83498030727422777627754279977, −1.33603645584750485578570703037, −0.55123189615188637671397130759, 1.726511213047655942687077112397, 2.48432638912638130572798140123, 3.01065439947354739184098295382, 3.6926730507313197717971046814, 4.504646323577496544440088184591, 4.86884872111621167732905941261, 6.00084394715797853208376717663, 6.57999164297467136533891969432, 7.24999322283088811306714791783, 8.21297827495892218259025052951, 8.54829111313020116626708430595, 9.123014938003500399338219272194, 10.40837425925892280468403572646, 10.97202236435994501646522464720, 11.41687423229215189113745861420, 12.24243342363220591093185718931, 13.117532351332548867450076611794, 13.54964689467939257647421182710, 14.454182757943803441110126105953, 14.83121960167901900557856859989, 15.41195703602368913114770486387, 15.73590069147036685741985520608, 16.4034722900363832480561272085, 17.35298419815219525224451996211, 18.11348992688913002400526762091

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