Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $0.843 - 0.537i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.733 − 0.680i)2-s + (0.900 + 0.433i)3-s + (0.0747 + 0.997i)4-s + (−0.365 + 0.930i)5-s + (−0.365 − 0.930i)6-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.900 − 0.433i)10-s + (−0.365 + 0.930i)12-s + (−0.955 − 0.294i)13-s + (−0.733 + 0.680i)14-s + (−0.733 + 0.680i)15-s + (−0.988 + 0.149i)16-s + (−0.988 + 0.149i)17-s + (0.0747 − 0.997i)18-s + ⋯
L(s,χ)  = 1  + (−0.733 − 0.680i)2-s + (0.900 + 0.433i)3-s + (0.0747 + 0.997i)4-s + (−0.365 + 0.930i)5-s + (−0.365 − 0.930i)6-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.900 − 0.433i)10-s + (−0.365 + 0.930i)12-s + (−0.955 − 0.294i)13-s + (−0.733 + 0.680i)14-s + (−0.733 + 0.680i)15-s + (−0.988 + 0.149i)16-s + (−0.988 + 0.149i)17-s + (0.0747 − 0.997i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.843 - 0.537i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.843 - 0.537i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $0.843 - 0.537i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (120, \cdot )$
Sato-Tate  :  $\mu(42)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ 0.843 - 0.537i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.320184258 - 0.3848648986i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.320184258 - 0.3848648986i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9157486953 - 0.08644162089i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9157486953 - 0.08644162089i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.83275012099368962803612188126, −17.118301442699972108025400642292, −16.467430693724916717447257318509, −15.633566224746277309855872333894, −15.29865538390737188132719826029, −14.74928188147456891235138304337, −13.9215470510224636803013910632, −13.27154153132355327140205835602, −12.60604589859079571553685115423, −11.73163298822630754149098351467, −11.40996233428697409729585753105, −9.941207481789299073702162981482, −9.55848888233056590836212201422, −8.93536435336221182763245731250, −8.5190969633365454922745827157, −7.6278261200156369478035066166, −7.46040414497647615812442799432, −6.361455574355370353928988576684, −5.737194871771362594808719716335, −4.84340693056395064795761235624, −4.31750533631255199028337628577, −3.08671894089235779402943558885, −2.235566687117027279609127958707, −1.65067296496666886989915826889, −0.73272753096375817482945021031, 0.5256725812301266392683567841, 1.64657555376631153589580706299, 2.59661586641007691549647169335, 2.92871572919812395438200725968, 3.734935592400394116193990150117, 4.35999138751057709655122241617, 5.03892310151546137169380990965, 6.833490860585561908291380971997, 6.965510443923473649059588981095, 7.71386621186852040383712860074, 8.328273785207714539665900127429, 9.07118323695255114149943151192, 9.871509742313218830137076895332, 10.23026947973036455429701436950, 10.98155946353351493022744888205, 11.30801806343672476794409303565, 12.34357758317251638187510887642, 13.14369804362592722118442710546, 13.67205470707507654418463231054, 14.360714561046574001911687602336, 15.13845472754804238596280646067, 15.57136263359584870739387165331, 16.532519961163926650353127159082, 16.96036014610571394424849846264, 17.903051863079347726677211840692

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