Properties

Degree 1
Conductor $ 3^{2} \cdot 11 $
Sign $0.342 + 0.939i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s − 10-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − 19-s + (0.5 − 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 26-s + ⋯
L(s,χ)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s − 10-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − 19-s + (0.5 − 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.342 + 0.939i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.342 + 0.939i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $0.342 + 0.939i$
motivic weight  =  \(0\)
character  :  $\chi_{99} (32, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 99,\ (0:\ ),\ 0.342 + 0.939i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7114354947 + 0.4981524963i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7114354947 + 0.4981524963i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8148834573 + 0.3814039884i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8148834573 + 0.3814039884i\)

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

−29.73311026503316888645237521567, −28.511004547828461118238724135768, −28.01032921274703323478547818296, −27.075255588305057313792802637180, −25.50520659460191880615914495354, −25.00060652514239518004593500373, −23.45096898482929298350645799793, −22.09924838819708722341315960882, −21.09041401137618106342581400656, −20.56577672808417917119819381912, −19.203996613214099341653821731530, −18.17851690481460124758534684615, −17.27248211087698777830560527031, −16.198978451997647766003481949269, −14.60809393450394363624237537361, −13.06437268624552902308022436243, −12.4267763582876905253253656489, −11.16098236568774588777796745337, −9.91486224309373574533049786584, −8.77847635966066878569064458651, −8.0148494082748976407028715759, −5.79390385693861197079072681815, −4.515034346670144754860877074409, −2.739500164748515075765218909709, −1.33431798093472058628095084409, 1.66111610522805616379320785197, 3.93261266859087398981281365932, 5.55127006670665499231414022906, 6.78128709607474239997558543134, 7.65792373074251517264811384482, 9.131013638035364790050205070407, 10.32101910268802337168169348222, 11.18824795574279542473790244499, 13.372944431725085989610424991478, 14.259151237161110063542761727134, 15.05452437546675296996801965152, 16.57764293398832160408901157153, 17.31338615090604738905826128760, 18.43922061588955605721425008512, 19.22351385890523347352222963149, 20.741775419317828356909066069162, 21.980887244633145961593449592265, 23.324103241242882074824831931637, 23.80858452583650324793712186753, 25.375164895319315797756007839848, 25.94019375437309691523494929062, 26.951541774435975137458661586228, 27.791300876198571010103663694826, 29.16827450804960318898788345848, 30.08423753099817190459794218465

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