Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $-0.862 - 0.505i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.899 + 0.435i)2-s + (0.809 + 0.587i)3-s + (0.619 + 0.784i)4-s + (−0.594 + 0.804i)5-s + (0.471 + 0.881i)6-s + (0.339 + 0.940i)7-s + (0.215 + 0.976i)8-s + (0.309 + 0.951i)9-s + (−0.885 + 0.464i)10-s + (0.0402 + 0.999i)12-s + (0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.953 + 0.301i)15-s + (−0.231 + 0.972i)16-s + (−0.932 + 0.362i)17-s + (−0.136 + 0.990i)18-s + ⋯
L(s,χ)  = 1  + (0.899 + 0.435i)2-s + (0.809 + 0.587i)3-s + (0.619 + 0.784i)4-s + (−0.594 + 0.804i)5-s + (0.471 + 0.881i)6-s + (0.339 + 0.940i)7-s + (0.215 + 0.976i)8-s + (0.309 + 0.951i)9-s + (−0.885 + 0.464i)10-s + (0.0402 + 0.999i)12-s + (0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.953 + 0.301i)15-s + (−0.231 + 0.972i)16-s + (−0.932 + 0.362i)17-s + (−0.136 + 0.990i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.862 - 0.505i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.862 - 0.505i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $-0.862 - 0.505i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (348, \cdot )$
Sato-Tate  :  $\mu(390)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ -0.862 - 0.505i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.9955625350 + 3.669346411i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.9955625350 + 3.669346411i\)
\(L(\chi,1)\)  \(\approx\)  \(1.290074782 + 1.712317124i\)
\(L(1,\chi)\)  \(\approx\)  \(1.290074782 + 1.712317124i\)

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

−17.3540385807598042400087120773, −16.51310518012820104598241709495, −15.72649978069885456481412222978, −15.32235374378975273368952840188, −14.5426399628290436134281266836, −13.83887124630157244580848780516, −13.31529282341664959435194232201, −12.99573311656515472829660519972, −12.14194687621309617210462534094, −11.61110719854856193662438350460, −10.83139129432411346777849991902, −10.21799356192548625284676979963, −9.2197199586620252303382887323, −8.570152873055112722951988426627, −7.94531745711945914718400871790, −7.14412559319643262550556471778, −6.60025850998098368701168664202, −5.757133314605397295500064303127, −4.65701368074612298996065596859, −4.32647580301088331166716975905, −3.53802443738592302886191205855, −2.99601764050996846940306847426, −1.83498030727422777627754279977, −1.33603645584750485578570703037, −0.55123189615188637671397130759, 1.726511213047655942687077112397, 2.48432638912638130572798140123, 3.01065439947354739184098295382, 3.6926730507313197717971046814, 4.504646323577496544440088184591, 4.86884872111621167732905941261, 6.00084394715797853208376717663, 6.57999164297467136533891969432, 7.24999322283088811306714791783, 8.21297827495892218259025052951, 8.54829111313020116626708430595, 9.123014938003500399338219272194, 10.40837425925892280468403572646, 10.97202236435994501646522464720, 11.41687423229215189113745861420, 12.24243342363220591093185718931, 13.117532351332548867450076611794, 13.54964689467939257647421182710, 14.454182757943803441110126105953, 14.83121960167901900557856859989, 15.41195703602368913114770486387, 15.73590069147036685741985520608, 16.4034722900363832480561272085, 17.35298419815219525224451996211, 18.11348992688913002400526762091

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