Properties

Degree 1
Conductor $ 3^{2} \cdot 11 $
Sign $-0.788 - 0.615i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.669 − 0.743i)2-s + (−0.104 + 0.994i)4-s + (0.669 − 0.743i)5-s + (−0.913 + 0.406i)7-s + (0.809 − 0.587i)8-s − 10-s + (0.978 − 0.207i)13-s + (0.913 + 0.406i)14-s + (−0.978 − 0.207i)16-s + (−0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + (0.669 + 0.743i)20-s + (−0.5 − 0.866i)23-s + (−0.104 − 0.994i)25-s + (−0.809 − 0.587i)26-s + ⋯
L(s,χ)  = 1  + (−0.669 − 0.743i)2-s + (−0.104 + 0.994i)4-s + (0.669 − 0.743i)5-s + (−0.913 + 0.406i)7-s + (0.809 − 0.587i)8-s − 10-s + (0.978 − 0.207i)13-s + (0.913 + 0.406i)14-s + (−0.978 − 0.207i)16-s + (−0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + (0.669 + 0.743i)20-s + (−0.5 − 0.866i)23-s + (−0.104 − 0.994i)25-s + (−0.809 − 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.788 - 0.615i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.788 - 0.615i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $-0.788 - 0.615i$
motivic weight  =  \(0\)
character  :  $\chi_{99} (85, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 99,\ (1:\ ),\ -0.788 - 0.615i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.3001865811 - 0.8722012604i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.3001865811 - 0.8722012604i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6458144472 - 0.4027560033i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6458144472 - 0.4027560033i\)

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

−29.85972447870807997620485747163, −29.00472709966800094872934560855, −28.04847487035792942712202071647, −26.68280972792627096109949593669, −25.96937392509166786074928695335, −25.36617953220345706992072131047, −23.99577284499026091961687854367, −22.96410489240109853112003479484, −22.05303285227343688412607279289, −20.47546184688558023975180402936, −19.2375677051744941925602571778, −18.414834870954978610322738384452, −17.37948201919605242347689234419, −16.34023445752799180026175015548, −15.29871881840874988054877473552, −14.06919564015423899138911120319, −13.21757872655044870959381995975, −11.12717344530329304446105755790, −10.10999947075686121870385198502, −9.23657572132016315605515510693, −7.69769640792034510121853206651, −6.50623284356052233853508477017, −5.74823641304121540663430676083, −3.610854371745213057488989642334, −1.61778106026359062133507829442, 0.51559900842817742622569221997, 2.154779542821513101348835821746, 3.59231606769337777899477208808, 5.36754681835059490472243716532, 6.9474085177468950611458456268, 8.67595489956514538401695387138, 9.31817355986088969281097925748, 10.45428075600824413906715855432, 11.85391236223301677670221636597, 12.91096182928154362002767080586, 13.69729218957009064390802662782, 15.881807419791988960768987188297, 16.54532578433025045903790043606, 17.892199684472815219466832060958, 18.620286221031763070459974080638, 20.02743083194564966013791255271, 20.636497995388402998504690709663, 21.859517284393921613482825031357, 22.66516967393205806577333074930, 24.422596073213303641314649626908, 25.47757225691715558260421691262, 26.15135193364656524230995235028, 27.54265693024382437714575075880, 28.526417757846904720958584275470, 29.011882552390980724156865994113

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