Properties

Degree $1$
Conductor $6017$
Sign $-0.875 - 0.483i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.269 + 0.963i)2-s + (−0.473 + 0.880i)3-s + (−0.854 + 0.518i)4-s + (−0.992 + 0.123i)5-s + (−0.975 − 0.219i)6-s + (−0.991 + 0.130i)7-s + (−0.729 − 0.683i)8-s + (−0.550 − 0.834i)9-s + (−0.386 − 0.922i)10-s + (−0.0517 − 0.998i)12-s + (−0.858 − 0.512i)13-s + (−0.393 − 0.919i)14-s + (0.361 − 0.932i)15-s + (0.461 − 0.887i)16-s + (−0.824 − 0.565i)17-s + (0.655 − 0.755i)18-s + ⋯
L(s,χ)  = 1  + (0.269 + 0.963i)2-s + (−0.473 + 0.880i)3-s + (−0.854 + 0.518i)4-s + (−0.992 + 0.123i)5-s + (−0.975 − 0.219i)6-s + (−0.991 + 0.130i)7-s + (−0.729 − 0.683i)8-s + (−0.550 − 0.834i)9-s + (−0.386 − 0.922i)10-s + (−0.0517 − 0.998i)12-s + (−0.858 − 0.512i)13-s + (−0.393 − 0.919i)14-s + (0.361 − 0.932i)15-s + (0.461 − 0.887i)16-s + (−0.824 − 0.565i)17-s + (0.655 − 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.875 - 0.483i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.875 - 0.483i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.875 - 0.483i$
Motivic weight: \(0\)
Character: $\chi_{6017} (173, \cdot )$
Sato-Tate group: $\mu(910)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.875 - 0.483i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.01768298641 + 0.06861134883i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.01768298641 + 0.06861134883i\)
\(L(\chi,1)\) \(\approx\) \(0.3801545475 + 0.3049906057i\)
\(L(1,\chi)\) \(\approx\) \(0.3801545475 + 0.3049906057i\)

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

−17.42337630525284996965705091616, −16.81652481217848634564407029484, −16.121444219167341913674244221894, −15.21297087939275766376870511055, −14.660713799211413000489282381019, −13.7259306521890810138211010499, −13.0736705828604696136851212010, −12.72265992844944383303803767482, −12.095353520123653410278123379, −11.47657589741757876416661676147, −10.97174519175433966037001976450, −10.2581278265297676092739852115, −9.33461816796516231963645566369, −8.79199830125171944597477101614, −7.89611042847517841339256704748, −7.20274710256940305657543274403, −6.507085409032249607459621619530, −5.76881598562728914999191904169, −4.96051961303363736068982074982, −4.06567553520861590840155892711, −3.708200247273151188068596361000, −2.478927852437301879347071965295, −2.16775618440883428038788254815, −1.0046456352280254068596891562, −0.04870106351531688443818176783, 0.399948814963310136739840024819, 2.45834539026454136179445876518, 3.33440704722048599439771425694, 3.815015262280868963671600566570, 4.54925822772874083140234240083, 5.11427476311645139585298478544, 5.95597436838805475608564963649, 6.61498555360817165083182625334, 7.148135034351542088504870024643, 8.05126331612477421978689535968, 8.69796241613702168720712833004, 9.40105930141282944059778231940, 10.01095886220565329487250527972, 10.76136598878839017288403175574, 11.569411155748894343030620859686, 12.397832676763044881501036328212, 12.65630442542532161991475645390, 13.56127585821532388521905810758, 14.7144969401714377136838109603, 14.85191745855342436325113924193, 15.5046568280165392876430946605, 16.2606351592178759447975438250, 16.44456261217860849382020535139, 17.097645488301471335705610489118, 18.00933893620544740543075056623

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