Properties

Degree 1
Conductor 199
Sign $-0.120 - 0.992i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(199\)
\( \varepsilon \)  =  $-0.120 - 0.992i$
motivic weight  =  \(0\)
character  :  $\chi_{199} (14, \cdot )$
Sato-Tate  :  $\mu(99)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 199,\ (0:\ ),\ -0.120 - 0.992i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7700773539 - 0.8696391847i$
$L(\frac12,\chi)$  $\approx$  $0.7700773539 - 0.8696391847i$
$L(\chi,1)$  $\approx$  0.9500435848 - 0.5277526406i
$L(1,\chi)$  $\approx$  0.9500435848 - 0.5277526406i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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