Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $-0.590 + 0.807i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.998 − 0.0483i)2-s + (−0.809 − 0.587i)3-s + (0.995 + 0.0965i)4-s + (0.715 − 0.698i)5-s + (0.779 + 0.626i)6-s + (0.836 + 0.548i)7-s + (−0.989 − 0.144i)8-s + (0.309 + 0.951i)9-s + (−0.748 + 0.663i)10-s + (−0.748 − 0.663i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (−0.989 + 0.144i)15-s + (0.981 + 0.192i)16-s + (−0.0724 + 0.997i)17-s + (−0.262 − 0.964i)18-s + ⋯
L(s,χ)  = 1  + (−0.998 − 0.0483i)2-s + (−0.809 − 0.587i)3-s + (0.995 + 0.0965i)4-s + (0.715 − 0.698i)5-s + (0.779 + 0.626i)6-s + (0.836 + 0.548i)7-s + (−0.989 − 0.144i)8-s + (0.309 + 0.951i)9-s + (−0.748 + 0.663i)10-s + (−0.748 − 0.663i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (−0.989 + 0.144i)15-s + (0.981 + 0.192i)16-s + (−0.0724 + 0.997i)17-s + (−0.262 − 0.964i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.590 + 0.807i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.590 + 0.807i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $-0.590 + 0.807i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (840, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ -0.590 + 0.807i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.2395641932 + 0.4719707860i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.2395641932 + 0.4719707860i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6117328679 + 0.009710676950i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6117328679 + 0.009710676950i\)

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

−17.67574223635813581902830821216, −16.869722496471010349822940199117, −16.49560110253684119610501445167, −15.59930297311191845178627700510, −14.86372284721918329586387431515, −14.689992465060695983022699328, −13.5004255895213913940867686584, −12.84890280063198572361029917119, −11.77733488945069361969353700705, −11.147760654951520200202704945630, −10.89973148149716428961104823851, −10.18497767231691452112518109651, −9.71105330726189489398767957074, −8.94027828478565325891647154519, −8.16094536400404681327174744531, −7.29250319417749278246863432520, −6.7606216892325703035617842374, −6.09836995043148417830534916914, −5.342721878054509604660183359316, −4.716325125541771109816576427738, −3.61504592750896420099072448, −2.80709707445467107315936028557, −1.99883383285249227145217920083, −1.04036885409974778506250401980, −0.217812360956921747374903170145, 1.313089764320015980064049859826, 1.6570947246702969386542077720, 2.0194882254376901934875143794, 3.3242572022304256585931810384, 4.52968908966811875657585037811, 5.24601637604738860826106338918, 5.93692705303031475965997249112, 6.43242492409604490378876588843, 7.22939665241543187190246019978, 8.02095927055316576767737929016, 8.71941085330827843620324321839, 9.021434953790700781562388326519, 10.12981672936987910496981203137, 10.594717126870862389198090319243, 11.35897910405667874789814628296, 11.888295685809244415399743426723, 12.52443090805599381104446724711, 13.08200693900984221885566011339, 14.000795488214755404111832694942, 14.7570563653298972405965400439, 15.539298711490763323311531843497, 16.38972794943523229676042854435, 16.841137485166142093943887670014, 17.30311593989528594975019180038, 17.87689860682642333843303450212

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