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Properties

Label 1-6017-6017.14-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.737 + 0.675i$
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  + (0.998 − 0.0598i)2-s + (0.753 + 0.657i)3-s + (0.992 − 0.119i)4-s + (−0.280 + 0.959i)5-s + (0.791 + 0.611i)6-s + (0.193 + 0.981i)7-s + (0.983 − 0.178i)8-s + (0.134 + 0.990i)9-s + (−0.222 + 0.974i)10-s + (0.826 + 0.563i)12-s + (−0.163 + 0.986i)13-s + (0.251 + 0.967i)14-s + (−0.842 + 0.538i)15-s + (0.971 − 0.237i)16-s + (0.971 − 0.237i)17-s + (0.193 + 0.981i)18-s + ⋯
L(s)  = 1  + (0.998 − 0.0598i)2-s + (0.753 + 0.657i)3-s + (0.992 − 0.119i)4-s + (−0.280 + 0.959i)5-s + (0.791 + 0.611i)6-s + (0.193 + 0.981i)7-s + (0.983 − 0.178i)8-s + (0.134 + 0.990i)9-s + (−0.222 + 0.974i)10-s + (0.826 + 0.563i)12-s + (−0.163 + 0.986i)13-s + (0.251 + 0.967i)14-s + (−0.842 + 0.538i)15-s + (0.971 − 0.237i)16-s + (0.971 − 0.237i)17-s + (0.193 + 0.981i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.737 + 0.675i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.737 + 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.881209696 + 4.838804454i\)
\(L(\frac12)\) \(\approx\) \(1.881209696 + 4.838804454i\)
\(L(1)\) \(\approx\) \(2.137865882 + 1.468588619i\)
\(L(1)\) \(\approx\) \(2.137865882 + 1.468588619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.998 - 0.0598i)T \)
3 \( 1 + (0.753 + 0.657i)T \)
5 \( 1 + (-0.280 + 0.959i)T \)
7 \( 1 + (0.193 + 0.981i)T \)
13 \( 1 + (-0.163 + 0.986i)T \)
17 \( 1 + (0.971 - 0.237i)T \)
19 \( 1 + (0.251 + 0.967i)T \)
23 \( 1 + (0.365 - 0.930i)T \)
29 \( 1 + (0.473 + 0.880i)T \)
31 \( 1 + (-0.995 - 0.0896i)T \)
37 \( 1 + (-0.646 + 0.762i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.733 - 0.680i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (-0.163 + 0.986i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (0.971 - 0.237i)T \)
67 \( 1 + (0.0747 - 0.997i)T \)
71 \( 1 + (0.971 - 0.237i)T \)
73 \( 1 + (0.251 - 0.967i)T \)
79 \( 1 + (0.858 + 0.512i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.772 - 0.635i)T \)
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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

−17.43413864155686449211688607263, −16.70831707235444102898961261625, −16.02489389457197984693259089514, −15.29327455357580966041959852030, −14.7948645210526809439468851855, −14.0057382613389669966814149498, −13.46702288074715851110647011278, −12.94238154657069860553184865894, −12.547212212716586074799302457184, −11.67843539839966454240478937106, −11.17683044206444831488887361585, −10.09163131112395160123429622414, −9.56205253056299471180459365380, −8.35697285296429000716543824734, −8.05163215617289581403385605309, −7.24815642942553175476635941056, −6.92590786480897478136976997488, −5.69094264673031839504782439096, −5.26890968657274676291417165763, −4.34340045202672266089146271579, −3.67965533542410440516412645980, −3.16075417608503126161439361561, −2.20593877410092763515833752296, −1.27913769891533664495515129536, −0.7731480989946601560263451404, 1.62292521149337293380676050794, 2.2262725272440628689740590664, 2.93178129471531849201663759586, 3.51232281761864773065752604784, 4.10472899639964014612794628481, 5.05533631901467033145689984440, 5.50753400465226028491841094911, 6.47884642250882590955056460160, 7.11445402069897303338022966736, 7.844947710490161331508520782282, 8.556754131221432636832466881992, 9.40348084951331770409094268126, 10.17578699097532915123949962719, 10.718715636956856230504449406614, 11.41726787854445418876840361807, 12.1642949461028360391713257459, 12.53783572390918085105963087565, 13.76227077607131379380352850756, 14.24047809918358919023317566485, 14.52957387858040313211234611639, 15.21084894785783752535675564776, 15.706515408470423168538950651917, 16.41206689584539577164333923580, 16.91947897555867335584543368860, 18.321405632164270557102115155386

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