Properties

Label 1-199-199.36-r0-0-0
Degree $1$
Conductor $199$
Sign $0.382 + 0.924i$
Analytic cond. $0.924152$
Root an. cond. $0.924152$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.916 − 0.400i)2-s + (0.873 + 0.486i)3-s + (0.678 + 0.734i)4-s + (0.580 + 0.814i)5-s + (−0.605 − 0.795i)6-s + (−0.987 + 0.158i)7-s + (−0.327 − 0.945i)8-s + (0.527 + 0.849i)9-s + (−0.204 − 0.978i)10-s + (−0.142 + 0.989i)11-s + (0.235 + 0.971i)12-s + (0.805 − 0.592i)13-s + (0.967 + 0.251i)14-s + (0.110 + 0.993i)15-s + (−0.0792 + 0.996i)16-s + (0.928 − 0.371i)17-s + ⋯
L(s)  = 1  + (−0.916 − 0.400i)2-s + (0.873 + 0.486i)3-s + (0.678 + 0.734i)4-s + (0.580 + 0.814i)5-s + (−0.605 − 0.795i)6-s + (−0.987 + 0.158i)7-s + (−0.327 − 0.945i)8-s + (0.527 + 0.849i)9-s + (−0.204 − 0.978i)10-s + (−0.142 + 0.989i)11-s + (0.235 + 0.971i)12-s + (0.805 − 0.592i)13-s + (0.967 + 0.251i)14-s + (0.110 + 0.993i)15-s + (−0.0792 + 0.996i)16-s + (0.928 − 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.382 + 0.924i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.382 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8423239420 + 0.5630324879i\)
\(L(\frac12)\) \(\approx\) \(0.8423239420 + 0.5630324879i\)
\(L(1)\) \(\approx\) \(0.9079699616 + 0.2585748270i\)
\(L(1)\) \(\approx\) \(0.9079699616 + 0.2585748270i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.916 - 0.400i)T \)
3 \( 1 + (0.873 + 0.486i)T \)
5 \( 1 + (0.580 + 0.814i)T \)
7 \( 1 + (-0.987 + 0.158i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (0.805 - 0.592i)T \)
17 \( 1 + (0.928 - 0.371i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.975 - 0.220i)T \)
29 \( 1 + (-0.553 - 0.832i)T \)
31 \( 1 + (-0.0158 + 0.999i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.999 + 0.0317i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.630 + 0.776i)T \)
53 \( 1 + (0.110 - 0.993i)T \)
59 \( 1 + (0.0475 - 0.998i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (0.967 - 0.251i)T \)
73 \( 1 + (0.296 + 0.954i)T \)
79 \( 1 + (0.902 - 0.429i)T \)
83 \( 1 + (0.235 - 0.971i)T \)
89 \( 1 + (0.997 - 0.0634i)T \)
97 \( 1 + (-0.857 - 0.513i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.36431669976558114559685062161, −25.73046625356959032622142735810, −25.25396483699339843319648197140, −23.93360543959288356518822664832, −23.722105096727412472069168954238, −21.68295891558073610340624846, −20.71534562625664685138122618916, −19.8895989091755394074258979928, −18.97117737389131633545488290969, −18.385352670845507476453816854605, −16.94708578875332807925847736468, −16.36505718657019274990366831052, −15.29723119747320887518678627359, −13.97147975870319410141134195122, −13.29480754570195097912158439892, −12.10177545498422263211227849266, −10.50544946023259090159415916725, −9.44383130233042896485827532507, −8.776272099537765621139929838479, −7.90253427910508420995771528373, −6.51685689299857742587233273574, −5.816895723489503962351050013847, −3.723003156296727278451508613334, −2.238584901113876107236012217, −0.981103262885870899678338645285, 1.95797423215800911512830019760, 2.91203587579845155622365528091, 3.854677498227460208633815036111, 6.00953581160394033376468702943, 7.19146307748916560512122483745, 8.21963236180261244419026078053, 9.49234586862146381954930146001, 10.01613269763437358836762174201, 10.785270784591699703409069982024, 12.42336705889689133492891151227, 13.34400066056429830860670827448, 14.65186085849966980454766666637, 15.59322984325758452208584293159, 16.451364971111710418807156474286, 17.75490567045803741101229344637, 18.65155034913973854450872126462, 19.36878087459784528890165626354, 20.41104466823284277022670310857, 21.12952718209855243783523924157, 22.11396029913090647038227386533, 23.02410031933717352470480656810, 25.0410165675690758637306092429, 25.61853059143742309820313392386, 25.9430259352137670272499046293, 27.00408231660418266881000348504

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