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Properties

Label 1-6017-6017.5301-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.970 + 0.239i$
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  + (0.955 + 0.294i)2-s + (0.900 − 0.433i)3-s + (0.826 + 0.563i)4-s + (0.988 + 0.149i)5-s + (0.988 − 0.149i)6-s + (0.826 − 0.563i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (0.900 + 0.433i)10-s + (0.988 + 0.149i)12-s + (0.733 + 0.680i)13-s + (0.955 − 0.294i)14-s + (0.955 − 0.294i)15-s + (0.365 + 0.930i)16-s + (0.365 + 0.930i)17-s + (0.826 − 0.563i)18-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)2-s + (0.900 − 0.433i)3-s + (0.826 + 0.563i)4-s + (0.988 + 0.149i)5-s + (0.988 − 0.149i)6-s + (0.826 − 0.563i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (0.900 + 0.433i)10-s + (0.988 + 0.149i)12-s + (0.733 + 0.680i)13-s + (0.955 − 0.294i)14-s + (0.955 − 0.294i)15-s + (0.365 + 0.930i)16-s + (0.365 + 0.930i)17-s + (0.826 − 0.563i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.970 + 0.239i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (5301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.970 + 0.239i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.523078901 + 0.9133050853i\)
\(L(\frac12)\) \(\approx\) \(7.523078901 + 0.9133050853i\)
\(L(1)\) \(\approx\) \(3.417404564 + 0.2891319359i\)
\(L(1)\) \(\approx\) \(3.417404564 + 0.2891319359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.955 + 0.294i)T \)
3 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (0.826 - 0.563i)T \)
13 \( 1 + (0.733 + 0.680i)T \)
17 \( 1 + (0.365 + 0.930i)T \)
19 \( 1 + (-0.955 + 0.294i)T \)
23 \( 1 + (-0.955 + 0.294i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (0.900 - 0.433i)T \)
37 \( 1 + (-0.365 - 0.930i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.826 - 0.563i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.733 - 0.680i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + (0.365 + 0.930i)T \)
71 \( 1 + (-0.365 - 0.930i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (-0.900 - 0.433i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
97 \( 1 + (0.955 - 0.294i)T \)
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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

−17.734371019639365115971505992218, −16.93880725808947724214279110104, −15.965063763460954792419648608240, −15.657736578285099999081418332838, −14.85419039126754879359549582256, −14.20370805383012008782358865706, −14.00269042654462523937742055263, −13.13550069416755465181206026903, −12.69411946044740509621600661065, −11.84898012812977083273029566881, −10.982557164003939304047787450, −10.456388237516974197502697626829, −9.84275739667857023293733914236, −9.0395887029342150152608358881, −8.41873737433264761796316914081, −7.68527787940964761998530786604, −6.73007882003937964443381423091, −5.951842619905426146604189608554, −5.27774839210750549618840992468, −4.737815959021064859821589918604, −4.03678792314512831199124095399, −2.983138639659735113285930042630, −2.64263290341960295660426073262, −1.7748618505816107857027803875, −1.247074274099126065225168098331, 1.294108528543294553748990958181, 1.933329550594598867802939437294, 2.27152650053465779142222326211, 3.4647004623254024622584395365, 3.985464156562785489353302139541, 4.59990451573392879543081596162, 5.73779876253910173693092349947, 6.1839767823031087601453777703, 6.86220537820374705292126795777, 7.61421861990920359313247914896, 8.2985270890013604760487429722, 8.75060103428804813532312904645, 9.90959958644899320673138876414, 10.40568177334478100487494962373, 11.30186376127441332392522379354, 11.92785834311080482996307497513, 12.909968916545775201886360567846, 13.233611258175239057201945764715, 13.89052905645944890006958786072, 14.394472520720009577109535000052, 14.754050499369254784745161135286, 15.52773958144527506545432947478, 16.372426515068026104341253845678, 17.17399130288751659392917934076, 17.533509684344782173931872399269

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