Properties

Degree $1$
Conductor $6017$
Sign $-0.933 + 0.357i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.594 − 0.804i)2-s + (0.809 − 0.587i)3-s + (−0.293 − 0.955i)4-s + (0.726 − 0.686i)5-s + (0.00805 − 0.999i)6-s + (0.737 − 0.675i)7-s + (−0.943 − 0.331i)8-s + (0.309 − 0.951i)9-s + (−0.120 − 0.992i)10-s + (−0.799 − 0.600i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (0.184 − 0.982i)15-s + (−0.827 + 0.561i)16-s + (0.937 − 0.347i)17-s + (−0.581 − 0.813i)18-s + ⋯
L(s,χ)  = 1  + (0.594 − 0.804i)2-s + (0.809 − 0.587i)3-s + (−0.293 − 0.955i)4-s + (0.726 − 0.686i)5-s + (0.00805 − 0.999i)6-s + (0.737 − 0.675i)7-s + (−0.943 − 0.331i)8-s + (0.309 − 0.951i)9-s + (−0.120 − 0.992i)10-s + (−0.799 − 0.600i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (0.184 − 0.982i)15-s + (−0.827 + 0.561i)16-s + (0.937 − 0.347i)17-s + (−0.581 − 0.813i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.933 + 0.357i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.933 + 0.357i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.933 + 0.357i$
Motivic weight: \(0\)
Character: $\chi_{6017} (965, \cdot )$
Sato-Tate group: $\mu(390)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.933 + 0.357i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.8478433968 - 4.582195275i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.8478433968 - 4.582195275i\)
\(L(\chi,1)\) \(\approx\) \(1.167786821 - 1.991213522i\)
\(L(1,\chi)\) \(\approx\) \(1.167786821 - 1.991213522i\)

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

−18.191200262495951494473553427278, −17.21187321175973721095709841646, −16.70485991691503478777120171308, −15.79859745651145745734772113902, −15.45314009932162254689769940787, −14.58049197595673573900585596611, −14.32685198386623165311589346479, −13.87297401510892899750580262043, −13.06139649373389535452242322828, −12.366042109919148828567868887526, −11.46198144638585472067256467765, −10.80215758786536785244711582055, −9.93532403718794216934455849599, −9.35989729181279904948807993470, −8.50485942587013077436167622439, −8.093736417856609216479110896437, −7.456015867385739413815631685155, −6.446700519313451399327264616248, −5.75358819370926591497986485786, −5.39895890026820908924910020331, −4.34660846704394841392141355551, −3.71888946858718879830248583342, −3.115380587699900281768013124925, −2.183526234105848022957033637793, −1.66892356040345719480803168238, 0.8517678256495445365956206270, 1.17593605798157061693270255668, 2.00079041615048143591476551, 2.66445189646472537479278760630, 3.52702050045964653169713560516, 4.29201213564122773117293373928, 4.81666339436535424591956712089, 5.91785705968019489148547799699, 6.200657232246659083819384681526, 7.37783033540232727651056941259, 8.02959422214846374500625941159, 8.82284574705578896574702843258, 9.34245585988105797013352957302, 10.08302578946892904250763209402, 10.72657828394461097427008719033, 11.58531758187303701018226031227, 12.149088841190917785409670427684, 13.01424999156941977681201692798, 13.39417115639901180831731686560, 13.81480075020336288093970387858, 14.49723950613154768913520164461, 15.01517188680314468687947471485, 15.920602263088865435146540370967, 16.77814206715598992416499813947, 17.58563116937881202417112142637

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