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Properties

Label 1-199-199.56-r0-0-0
Degree $1$
Conductor $199$
Sign $0.901 + 0.433i$
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  + (−0.0792 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.901 + 0.433i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.901 + 0.433i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7086742668 + 0.1614002511i\)
\(L(\frac12)\) \(\approx\) \(0.7086742668 + 0.1614002511i\)
\(L(1)\) \(\approx\) \(0.6933837436 + 0.2910196252i\)
\(L(1)\) \(\approx\) \(0.6933837436 + 0.2910196252i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.0792 + 0.996i)T \)
3 \( 1 + (-0.444 + 0.895i)T \)
5 \( 1 + (-0.786 - 0.618i)T \)
7 \( 1 + (0.805 - 0.592i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (-0.823 - 0.567i)T \)
17 \( 1 + (0.0475 - 0.998i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.630 + 0.776i)T \)
29 \( 1 + (-0.701 - 0.712i)T \)
31 \( 1 + (0.997 + 0.0634i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (0.991 - 0.126i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.916 - 0.400i)T \)
53 \( 1 + (0.902 + 0.429i)T \)
59 \( 1 + (0.981 + 0.189i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (0.527 - 0.849i)T \)
73 \( 1 + (0.356 - 0.934i)T \)
79 \( 1 + (-0.204 - 0.978i)T \)
83 \( 1 + (0.580 + 0.814i)T \)
89 \( 1 + (0.967 - 0.251i)T \)
97 \( 1 + (-0.553 + 0.832i)T \)
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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

−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