Properties

Degree 1
Conductor 199
Sign $0.976 - 0.217i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.873 − 0.486i)2-s + (0.678 + 0.734i)3-s + (0.527 − 0.849i)4-s + (0.723 + 0.690i)5-s + (0.950 + 0.312i)6-s + (−0.605 − 0.795i)7-s + (0.0475 − 0.998i)8-s + (−0.0792 + 0.996i)9-s + (0.967 + 0.251i)10-s + (0.415 − 0.909i)11-s + (0.981 − 0.189i)12-s + (−0.857 − 0.513i)13-s + (−0.916 − 0.400i)14-s + (−0.0158 + 0.999i)15-s + (−0.444 − 0.895i)16-s + (0.580 + 0.814i)17-s + ⋯
L(s,χ)  = 1  + (0.873 − 0.486i)2-s + (0.678 + 0.734i)3-s + (0.527 − 0.849i)4-s + (0.723 + 0.690i)5-s + (0.950 + 0.312i)6-s + (−0.605 − 0.795i)7-s + (0.0475 − 0.998i)8-s + (−0.0792 + 0.996i)9-s + (0.967 + 0.251i)10-s + (0.415 − 0.909i)11-s + (0.981 − 0.189i)12-s + (−0.857 − 0.513i)13-s + (−0.916 − 0.400i)14-s + (−0.0158 + 0.999i)15-s + (−0.444 − 0.895i)16-s + (0.580 + 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.976 - 0.217i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.976 - 0.217i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(199\)
\( \varepsilon \)  =  $0.976 - 0.217i$
motivic weight  =  \(0\)
character  :  $\chi_{199} (46, \cdot )$
Sato-Tate  :  $\mu(99)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 199,\ (0:\ ),\ 0.976 - 0.217i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.380008338 - 0.2615733680i$
$L(\frac12,\chi)$  $\approx$  $2.380008338 - 0.2615733680i$
$L(\chi,1)$  $\approx$  2.026311632 - 0.1819203606i
$L(1,\chi)$  $\approx$  2.026311632 - 0.1819203606i

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

−26.382698085460857383583987326268, −25.62951333676884173486088816319, −24.90837216579454764932389981007, −24.43148498456305807144542045802, −23.33316617851990189385647616872, −22.20551860549005655389725966095, −21.39207282132464936903460619373, −20.33488996772476587364109947019, −19.605037954222523184139835109086, −18.15761882809749847437915401276, −17.3076468954507430545388980253, −16.174361465492920557020442949222, −15.11198457872172446332126230709, −14.15923136463820277506314114763, −13.38869169589178264658298160191, −12.29364923595393976571178693414, −12.049069580965363020511417174957, −9.622432265332348726638491041214, −8.95695404919164181502136937983, −7.580366655690814762543168200160, −6.65332219059251749221631099181, −5.61310154889105190395567175455, −4.40455247042631759286664869052, −2.79731479073541695332282604457, −1.967206743898896262275485915196, 1.857974753811733244019264470805, 3.256508945083684943742994793335, 3.717442183372147507553257965371, 5.30697079614342662860241446646, 6.31010593191671524633542250409, 7.652472819237306417080143347177, 9.38752481603141013319796114956, 10.31457048556781304155582562003, 10.69626991867707449695438251520, 12.33102293582336775002415656212, 13.52355408872763524454930602980, 14.20752699239221404896237993520, 14.80865991755816553937170574062, 16.09623803113433715075707198821, 16.97911870896824805929543399422, 18.71196563236992651565214834528, 19.560373163486532199783766732354, 20.31747264474376056753530617234, 21.38629979155931948209123923896, 22.063654568313618236094667103547, 22.675452882548708440876469754533, 23.97608085336962880373519987726, 25.091794019243023640683020534098, 25.867818594535895350603826537957, 26.86633754094975179002673838624

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