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Properties

Label 1-6017-6017.3065-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.385 + 0.922i$
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  + (0.937 − 0.347i)2-s + (0.809 + 0.587i)3-s + (0.759 − 0.650i)4-s + (−0.818 + 0.574i)5-s + (0.962 + 0.270i)6-s + (0.997 − 0.0643i)7-s + (0.485 − 0.873i)8-s + (0.309 + 0.951i)9-s + (−0.568 + 0.822i)10-s + (0.996 − 0.0804i)12-s + (−0.669 + 0.743i)13-s + (0.913 − 0.406i)14-s + (−0.999 − 0.0161i)15-s + (0.152 − 0.988i)16-s + (0.870 + 0.493i)17-s + (0.619 + 0.784i)18-s + ⋯
L(s)  = 1  + (0.937 − 0.347i)2-s + (0.809 + 0.587i)3-s + (0.759 − 0.650i)4-s + (−0.818 + 0.574i)5-s + (0.962 + 0.270i)6-s + (0.997 − 0.0643i)7-s + (0.485 − 0.873i)8-s + (0.309 + 0.951i)9-s + (−0.568 + 0.822i)10-s + (0.996 − 0.0804i)12-s + (−0.669 + 0.743i)13-s + (0.913 − 0.406i)14-s + (−0.999 − 0.0161i)15-s + (0.152 − 0.988i)16-s + (0.870 + 0.493i)17-s + (0.619 + 0.784i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.385 + 0.922i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (3065, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.385 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.886105128 + 2.586616571i\)
\(L(\frac12)\) \(\approx\) \(3.886105128 + 2.586616571i\)
\(L(1)\) \(\approx\) \(2.354219974 + 0.4729035907i\)
\(L(1)\) \(\approx\) \(2.354219974 + 0.4729035907i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.937 - 0.347i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.818 + 0.574i)T \)
7 \( 1 + (0.997 - 0.0643i)T \)
13 \( 1 + (-0.669 + 0.743i)T \)
17 \( 1 + (0.870 + 0.493i)T \)
19 \( 1 + (0.704 + 0.709i)T \)
23 \( 1 + (-0.692 + 0.721i)T \)
29 \( 1 + (-0.644 + 0.764i)T \)
31 \( 1 + (0.998 + 0.0483i)T \)
37 \( 1 + (0.984 + 0.176i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.428 - 0.903i)T \)
47 \( 1 + (-0.789 + 0.613i)T \)
53 \( 1 + (-0.657 - 0.753i)T \)
59 \( 1 + (0.704 - 0.709i)T \)
61 \( 1 + (-0.962 - 0.270i)T \)
67 \( 1 + (0.428 - 0.903i)T \)
71 \( 1 + (-0.541 - 0.840i)T \)
73 \( 1 + (0.293 - 0.955i)T \)
79 \( 1 + (-0.607 + 0.794i)T \)
83 \( 1 + (0.0884 + 0.996i)T \)
89 \( 1 + (-0.885 + 0.464i)T \)
97 \( 1 + (0.594 + 0.804i)T \)
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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

−17.51667689383458211718520318873, −16.8794642658817572463535726394, −15.98967898384291995316648425392, −15.474719038292363754715359244411, −14.82836657643751697600095787337, −14.38796085890586566609430995441, −13.72585406150522721229755601682, −12.99305235457326196291530260075, −12.473028380575625830325302323047, −11.64379456779055525885085328476, −11.62488636120950519845190590210, −10.361565166227836906073248004551, −9.43534210663547503727268660001, −8.50237695472708532263023816819, −8.023263685292104959960182580335, −7.53437630628658006979110612331, −7.09512242016489751203305302119, −5.93838629602931587218469142446, −5.297293346864803624986086864199, −4.477030182068190626519287043669, −4.05886854792325249830528284654, −2.97681403179973602894221770080, −2.61629831691460053827918482394, −1.564174670764234494114704305021, −0.70667207817417421243056751221, 1.32080640899449935805124808540, 1.95236690747360559194189951441, 2.82094089563171669100132912935, 3.50785266398802305631947522602, 4.01596814635423694162523323325, 4.72406123010667306183311043276, 5.275563971580512336899093123468, 6.26664374349638191270692765132, 7.20625778261586184592751159792, 7.84472373656071521474804082977, 8.14825515560301452994805849170, 9.47963558132145454586815569191, 9.913810648095854069959343874399, 10.73394084732638578143326287345, 11.26129452680403158057149444820, 11.91385489279391474309666436947, 12.431710266416333259898007186163, 13.48959869245501722736197890020, 14.21150965936300035178238700020, 14.44066027419714317228549217750, 14.9980912489279078624568841975, 15.582928627580661851382169122220, 16.33041494956571771549520129672, 16.82800903057438681652061794272, 18.07412818471844991527053355228

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