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Properties

Label 1-199-199.51-r0-0-0
Degree $1$
Conductor $199$
Sign $0.773 - 0.634i$
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  + (−0.745 + 0.666i)2-s + (−0.701 − 0.712i)3-s + (0.110 − 0.993i)4-s + (−0.888 + 0.458i)5-s + (0.997 + 0.0634i)6-s + (0.902 − 0.429i)7-s + (0.580 + 0.814i)8-s + (−0.0158 + 0.999i)9-s + (0.356 − 0.934i)10-s + (−0.654 + 0.755i)11-s + (−0.786 + 0.618i)12-s + (−0.204 − 0.978i)13-s + (−0.386 + 0.922i)14-s + (0.950 + 0.312i)15-s + (−0.975 − 0.220i)16-s + (0.981 + 0.189i)17-s + ⋯
L(s)  = 1  + (−0.745 + 0.666i)2-s + (−0.701 − 0.712i)3-s + (0.110 − 0.993i)4-s + (−0.888 + 0.458i)5-s + (0.997 + 0.0634i)6-s + (0.902 − 0.429i)7-s + (0.580 + 0.814i)8-s + (−0.0158 + 0.999i)9-s + (0.356 − 0.934i)10-s + (−0.654 + 0.755i)11-s + (−0.786 + 0.618i)12-s + (−0.204 − 0.978i)13-s + (−0.386 + 0.922i)14-s + (0.950 + 0.312i)15-s + (−0.975 − 0.220i)16-s + (0.981 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.773 - 0.634i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.773 - 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4763930819 - 0.1703909080i\)
\(L(\frac12)\) \(\approx\) \(0.4763930819 - 0.1703909080i\)
\(L(1)\) \(\approx\) \(0.5513945579 + 0.01737194956i\)
\(L(1)\) \(\approx\) \(0.5513945579 + 0.01737194956i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.745 + 0.666i)T \)
3 \( 1 + (-0.701 - 0.712i)T \)
5 \( 1 + (-0.888 + 0.458i)T \)
7 \( 1 + (0.902 - 0.429i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (-0.204 - 0.978i)T \)
17 \( 1 + (0.981 + 0.189i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.805 - 0.592i)T \)
29 \( 1 + (0.527 - 0.849i)T \)
31 \( 1 + (-0.266 - 0.963i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.857 - 0.513i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.823 - 0.567i)T \)
53 \( 1 + (0.950 - 0.312i)T \)
59 \( 1 + (0.723 + 0.690i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (-0.386 - 0.922i)T \)
73 \( 1 + (-0.916 - 0.400i)T \)
79 \( 1 + (0.296 + 0.954i)T \)
83 \( 1 + (-0.786 - 0.618i)T \)
89 \( 1 + (0.472 + 0.881i)T \)
97 \( 1 + (0.967 + 0.251i)T \)
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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

−27.291457498587309938982320511868, −26.60961181794174497116757614882, −25.365919801081427271936560831610, −23.92566537413147206667582240864, −23.36833571079897178332938868064, −21.79887330626139874059605905511, −21.30727946706637154189399977962, −20.54738447414489603539621983509, −19.26968302824626654115823710951, −18.503455687134768979405061704606, −17.36450206371989198352038252093, −16.514823270687261927755116808384, −15.822029656181969774585585538782, −14.626784409067128350798102020697, −12.85684048768813842179069952763, −11.7900579204552727047186996213, −11.33749386733209315253183209748, −10.37619203817880660145430279486, −9.0161065349287045184052039178, −8.35594190606113215624527949086, −7.05814610308748602043136003434, −5.25054689279056965248187695804, −4.31765679404940201454787903392, −3.07159119292465108997717990042, −1.17175831657843318632949316299, 0.66214592182541014240362877045, 2.25568338732499168552613956712, 4.51766746843459908357338998935, 5.57872943574643798504866026824, 6.87027630161626100553717906192, 7.78070220863552539029987491453, 8.14387454212741551964135321720, 10.24458491149991481776993277474, 10.8048792720407674558846942856, 11.87890586501647377403024007811, 13.07175908175754901697975933636, 14.58194488308136400889195755622, 15.17242346575841534478193858914, 16.45204142973541895979845751492, 17.31541553131873006441036184311, 18.11427680373060718929052358289, 18.86761683911471412246555613491, 19.78344053867607268049460535651, 20.89467489142592996848449472102, 22.78145393069488022843708836967, 23.1934859592911541274118547184, 23.94505776178463576349970015837, 24.885578294458888497571841446259, 25.82521103222124656415709757718, 27.02839152693761927109766527569

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