Properties

Degree $1$
Conductor $6017$
Sign $0.385 + 0.922i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.385 + 0.922i$
Motivic weight: \(0\)
Character: $\chi_{6017} (3065, \cdot )$
Sato-Tate group: $\mu(390)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.385 + 0.922i)\)

Particular Values

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

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

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