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Properties

Label 1-199-199.14-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.120 - 0.992i$
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  + (0.110 − 0.993i)2-s + (−0.0158 + 0.999i)3-s + (−0.975 − 0.220i)4-s + (0.580 − 0.814i)5-s + (0.991 + 0.126i)6-s + (0.630 − 0.776i)7-s + (−0.327 + 0.945i)8-s + (−0.999 − 0.0317i)9-s + (−0.745 − 0.666i)10-s + (−0.142 − 0.989i)11-s + (0.235 − 0.971i)12-s + (−0.916 + 0.400i)13-s + (−0.701 − 0.712i)14-s + (0.805 + 0.592i)15-s + (0.902 + 0.429i)16-s + (0.928 + 0.371i)17-s + ⋯
L(s)  = 1  + (0.110 − 0.993i)2-s + (−0.0158 + 0.999i)3-s + (−0.975 − 0.220i)4-s + (0.580 − 0.814i)5-s + (0.991 + 0.126i)6-s + (0.630 − 0.776i)7-s + (−0.327 + 0.945i)8-s + (−0.999 − 0.0317i)9-s + (−0.745 − 0.666i)10-s + (−0.142 − 0.989i)11-s + (0.235 − 0.971i)12-s + (−0.916 + 0.400i)13-s + (−0.701 − 0.712i)14-s + (0.805 + 0.592i)15-s + (0.902 + 0.429i)16-s + (0.928 + 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.120 - 0.992i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.120 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7700773539 - 0.8696391847i\)
\(L(\frac12)\) \(\approx\) \(0.7700773539 - 0.8696391847i\)
\(L(1)\) \(\approx\) \(0.9500435848 - 0.5277526406i\)
\(L(1)\) \(\approx\) \(0.9500435848 - 0.5277526406i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.110 - 0.993i)T \)
3 \( 1 + (-0.0158 + 0.999i)T \)
5 \( 1 + (0.580 - 0.814i)T \)
7 \( 1 + (0.630 - 0.776i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (-0.916 + 0.400i)T \)
17 \( 1 + (0.928 + 0.371i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (0.296 - 0.954i)T \)
29 \( 1 + (-0.444 - 0.895i)T \)
31 \( 1 + (-0.857 + 0.513i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.472 + 0.881i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.356 + 0.934i)T \)
53 \( 1 + (0.805 - 0.592i)T \)
59 \( 1 + (0.0475 + 0.998i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (-0.995 - 0.0950i)T \)
71 \( 1 + (-0.701 + 0.712i)T \)
73 \( 1 + (0.678 + 0.734i)T \)
79 \( 1 + (-0.823 + 0.567i)T \)
83 \( 1 + (0.235 + 0.971i)T \)
89 \( 1 + (-0.553 + 0.832i)T \)
97 \( 1 + (0.873 + 0.486i)T \)
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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

−27.04468324639037083270713369293, −25.62804694888637422614275732265, −25.39816846436822756117244372988, −24.50359676181129125072219532566, −23.54636105292146687417925954045, −22.59580173805999878318280692424, −21.940847337554983257046490385667, −20.56560409651901913192428244490, −19.01184815554730289858586996389, −18.28226218319949597475224668063, −17.725519817120536548554215302294, −16.84370624845078359543808545583, −15.18193689380735564395243538927, −14.62399187294882099985300987625, −13.7801424553940053164246433720, −12.6197814808829859402898202752, −11.79628966022256140847485337453, −10.081982380230602789912606631195, −9.01797957584254583933060894760, −7.46237373734089236957655483871, −7.3538129093994107042107415837, −5.76432002873023192257352666639, −5.25289202345695944886278822714, −3.13451716309985529110397978040, −1.79526040752372103213968648582, 0.94236967284914337027203314750, 2.60188009195233996339850033702, 3.96522941858092164718966380685, 4.86722469274319531739234211442, 5.69143798523394380790476543692, 7.99550109832200473054550225129, 9.06492723313976494735782157182, 9.8604202840485654636402772387, 10.80548587446230377533566660434, 11.67350976173371921372241888209, 12.92199971779549405583977559298, 14.04323829338998197286427032701, 14.587299375032651261642893762, 16.37282029423755680797218563800, 16.99134457242612126158660624084, 17.946798911486373249949624669951, 19.4366235541214630919063511440, 20.22253964159231525683952406309, 21.15755007602951981491762897495, 21.474231408922711710324838826062, 22.56551793838201739913684407885, 23.715898829528810478860661460967, 24.57470779441180390791896170971, 26.24847123404863620174934464248, 26.825143311048959455737395292187

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