Properties

Degree $1$
Conductor $6017$
Sign $-0.372 + 0.927i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.657 + 0.753i)2-s + (0.809 − 0.587i)3-s + (−0.136 − 0.990i)4-s + (−0.457 − 0.889i)5-s + (−0.0884 + 0.996i)6-s + (−0.293 − 0.955i)7-s + (0.836 + 0.548i)8-s + (0.309 − 0.951i)9-s + (0.970 + 0.239i)10-s + (−0.692 − 0.721i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (−0.892 − 0.450i)15-s + (−0.962 + 0.270i)16-s + (−0.726 + 0.686i)17-s + (0.513 + 0.857i)18-s + ⋯
L(s,χ)  = 1  + (−0.657 + 0.753i)2-s + (0.809 − 0.587i)3-s + (−0.136 − 0.990i)4-s + (−0.457 − 0.889i)5-s + (−0.0884 + 0.996i)6-s + (−0.293 − 0.955i)7-s + (0.836 + 0.548i)8-s + (0.309 − 0.951i)9-s + (0.970 + 0.239i)10-s + (−0.692 − 0.721i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (−0.892 − 0.450i)15-s + (−0.962 + 0.270i)16-s + (−0.726 + 0.686i)17-s + (0.513 + 0.857i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.372 + 0.927i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.372 + 0.927i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.372 + 0.927i$
Motivic weight: \(0\)
Character: $\chi_{6017} (1196, \cdot )$
Sato-Tate group: $\mu(390)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.372 + 0.927i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.2062596657 - 0.3051085249i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.2062596657 - 0.3051085249i\)
\(L(\chi,1)\) \(\approx\) \(0.6457011702 - 0.2749300914i\)
\(L(1,\chi)\) \(\approx\) \(0.6457011702 - 0.2749300914i\)

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

−18.50077480974519710201202008756, −17.72602741415044807935245746275, −16.74997334666534204892959553675, −16.25577781013262418128112000655, −15.47357507488058216037241159877, −14.97443525849346929233404470387, −14.36300450875560735113643984530, −13.45867379762635418759873634751, −12.92975036785455726902865141750, −11.96670331739610100875318821731, −11.39282027251794964159511947341, −11.00148608006564046631291042120, −9.86986568771900965823262342568, −9.67219977596967621222392148463, −9.06277523080073850825679678921, −8.163708595281911048707275989719, −7.7708322074163532525201283187, −6.9452061851580231523313840999, −6.17625624697205387869491412039, −4.90270655480868842850115570067, −4.27490577102927549286872089147, −3.462853025767079680016362969950, −2.93241170190291370473767145928, −2.2474464182165625128900237083, −1.73833212235805231480858991573, 0.130142158782712477821563221108, 0.744564478467084517663133738940, 1.662331418083663597137727157779, 2.44612647836246495120086132729, 3.594416674984564248014485475584, 4.31332568908342465952245893672, 4.96832147655695814591223944262, 5.93733338209466970679697247509, 6.89281631048565806083310707356, 7.10565792865504042944028036780, 8.10700842230875911203143213168, 8.26658501675635050730105740253, 9.20941595109909392266942543019, 9.52498680723585581689805666126, 10.63977058419949565379400452135, 10.958845044805965405637185017682, 12.32663262599188716041726434520, 12.78840785729487752237411430299, 13.34638063358492589080069620206, 14.07587434094711884983314221307, 14.746175334656219345099658281279, 15.39171810322299470078451777207, 15.86928144768149211831177924926, 16.76352402038106374003250624477, 17.273611093345665416950605079369

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