Properties

Degree 1
Conductor 199
Sign $0.901 + 0.433i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.0792 + 0.996i)2-s + (−0.444 + 0.895i)3-s + (−0.987 − 0.158i)4-s + (−0.786 − 0.618i)5-s + (−0.857 − 0.513i)6-s + (0.805 − 0.592i)7-s + (0.235 − 0.971i)8-s + (−0.605 − 0.795i)9-s + (0.678 − 0.734i)10-s + (0.841 + 0.540i)11-s + (0.580 − 0.814i)12-s + (−0.823 − 0.567i)13-s + (0.527 + 0.849i)14-s + (0.902 − 0.429i)15-s + (0.950 + 0.312i)16-s + (0.0475 − 0.998i)17-s + ⋯
L(s,χ)  = 1  + (−0.0792 + 0.996i)2-s + (−0.444 + 0.895i)3-s + (−0.987 − 0.158i)4-s + (−0.786 − 0.618i)5-s + (−0.857 − 0.513i)6-s + (0.805 − 0.592i)7-s + (0.235 − 0.971i)8-s + (−0.605 − 0.795i)9-s + (0.678 − 0.734i)10-s + (0.841 + 0.540i)11-s + (0.580 − 0.814i)12-s + (−0.823 − 0.567i)13-s + (0.527 + 0.849i)14-s + (0.902 − 0.429i)15-s + (0.950 + 0.312i)16-s + (0.0475 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.901 + 0.433i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.901 + 0.433i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(199\)
\( \varepsilon \)  =  $0.901 + 0.433i$
motivic weight  =  \(0\)
character  :  $\chi_{199} (56, \cdot )$
Sato-Tate  :  $\mu(99)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 199,\ (0:\ ),\ 0.901 + 0.433i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7086742668 + 0.1614002511i$
$L(\frac12,\chi)$  $\approx$  $0.7086742668 + 0.1614002511i$
$L(\chi,1)$  $\approx$  0.6933837436 + 0.2910196252i
$L(1,\chi)$  $\approx$  0.6933837436 + 0.2910196252i

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

−27.21258368064203314901387810800, −26.17411377595378049074461509705, −24.64351521925089971436249536002, −23.971082668443397954719539063733, −22.85662958853164951126158142836, −22.19759987180603513548441280697, −21.25024756983056017191082412289, −19.87133683737857188346457585855, −19.04674789256813235585564987663, −18.6373122756243992009024532161, −17.51786295217385141586408381187, −16.65254233679422028113834799336, −14.69882740885809840628751879134, −14.24260755031885774878475851545, −12.731443300962508938989667688904, −11.882624427503019701026754095367, −11.41288758472376587374076977853, −10.37970579475667504286054704630, −8.69965340620397854883153170729, −7.96637348483430118200377729877, −6.645046993835733684715898605125, −5.24905257689743763031176764408, −3.92448857841215943156109988043, −2.52703524452067291659609852531, −1.36610535589662192830175762303, 0.71124096845744992460112946648, 3.66709245593512768531704599954, 4.742256947111646443733688447202, 5.147268380289672694557306037556, 6.89725840869827329853400970687, 7.79238970608917011005519521613, 9.021505090334016024594649332146, 9.81155238514931898949286197619, 11.231218663429620330492914751633, 12.14547385366648332024556396443, 13.60108786899536727288671547274, 14.831913244853415742108450813568, 15.356537717439908034171067924591, 16.397290950896154998067840909427, 17.2538514673768931920531268659, 17.72650590551076792323658365117, 19.47167706141734441302143552557, 20.32240847562497574482613551683, 21.430659672697760217018519762930, 22.70927492643090213610226194922, 23.075791791515687120039692787336, 24.26186680955881776695084066872, 24.86711176405244429801913881833, 26.326172796668237216934317004385, 27.04297917942600815213806451818

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