Properties

Label 1-199-199.46-r0-0-0
Degree $1$
Conductor $199$
Sign $0.976 - 0.217i$
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  + (0.873 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.976 - 0.217i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.976 - 0.217i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.380008338 - 0.2615733680i\)
\(L(\frac12)\) \(\approx\) \(2.380008338 - 0.2615733680i\)
\(L(1)\) \(\approx\) \(2.026311632 - 0.1819203606i\)
\(L(1)\) \(\approx\) \(2.026311632 - 0.1819203606i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.873 - 0.486i)T \)
3 \( 1 + (0.678 + 0.734i)T \)
5 \( 1 + (0.723 + 0.690i)T \)
7 \( 1 + (-0.605 - 0.795i)T \)
11 \( 1 + (0.415 - 0.909i)T \)
13 \( 1 + (-0.857 - 0.513i)T \)
17 \( 1 + (0.580 + 0.814i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.999 + 0.0317i)T \)
29 \( 1 + (0.356 + 0.934i)T \)
31 \( 1 + (-0.975 - 0.220i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.902 - 0.429i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.991 - 0.126i)T \)
53 \( 1 + (-0.0158 - 0.999i)T \)
59 \( 1 + (-0.786 - 0.618i)T \)
61 \( 1 + (-0.959 - 0.281i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (-0.916 + 0.400i)T \)
73 \( 1 + (0.472 - 0.881i)T \)
79 \( 1 + (0.997 + 0.0634i)T \)
83 \( 1 + (0.981 + 0.189i)T \)
89 \( 1 + (0.630 - 0.776i)T \)
97 \( 1 + (0.296 + 0.954i)T \)
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

−26.382698085460857383583987326268, −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