↑

Properties

Label 1-199-199.128-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

Related objects

Downloads

Learn more

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: $\chi_{199} (128, \cdot )$
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 \( 1 + (0.873 - 0.486i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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