Properties

Label 1-199-199.103-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.887 + 0.459i$
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  + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.841 + 0.540i)5-s + (−0.959 + 0.281i)6-s + (0.415 − 0.909i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.654 + 0.755i)10-s + (−0.654 + 0.755i)11-s + (−0.142 − 0.989i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)15-s + (0.841 + 0.540i)16-s + (−0.654 + 0.755i)17-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.841 + 0.540i)5-s + (−0.959 + 0.281i)6-s + (0.415 − 0.909i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.654 + 0.755i)10-s + (−0.654 + 0.755i)11-s + (−0.142 − 0.989i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)15-s + (0.841 + 0.540i)16-s + (−0.654 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.887 + 0.459i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.887 + 0.459i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2867852231 + 1.177351152i\)
\(L(\frac12)\) \(\approx\) \(0.2867852231 + 1.177351152i\)
\(L(1)\) \(\approx\) \(0.7197069585 + 0.8548338677i\)
\(L(1)\) \(\approx\) \(0.7197069585 + 0.8548338677i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.142 + 0.989i)T \)
3 \( 1 + (0.415 + 0.909i)T \)
5 \( 1 + (0.841 + 0.540i)T \)
7 \( 1 + (0.415 - 0.909i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.959 + 0.281i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (-0.142 - 0.989i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (-0.959 + 0.281i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (-0.142 + 0.989i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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

−26.57045796565711974779324150889, −25.490919293308973846680944765443, −24.66164202592614423933327664254, −23.86836633783299670867956814049, −22.43421500781274257398398742699, −21.64634793888655290534778525656, −20.55278385648077663593708438502, −20.07788828732371469807742405974, −18.71205639387235989900473728536, −18.09295619414521049602781501341, −17.53109332447425027618436051743, −15.93243988552975558193420694262, −14.30810093176592429701447901670, −13.61878530423315922233078504560, −12.7024182140661266474065663667, −11.9963158760124933555199396192, −10.76775306519989994023843846841, −9.420871943457484304333893773142, −8.65197024887698438385290408450, −7.784510854457659505169958810464, −5.87504730529765393799533949174, −5.05291426081244463496050874871, −3.01831979411871509323386540113, −2.28927863145751553742356674732, −0.99576459872401436335208643376, 2.07739978545851977617590186897, 3.91091119975625763412285474068, 4.77989991462198609287259926909, 5.99352385948400964023070259326, 7.23765556891576182474643965893, 8.19046974449786343163744704630, 9.65564163493132214437423837467, 9.9694139125927713155118727621, 11.1885612677874537800224090546, 13.31916719964140199814737593164, 13.94654467940508123295574552210, 14.76263218397908503025397264744, 15.63524932829510305894280659127, 16.76495695966973847846509864882, 17.45652852486170623930017197459, 18.41429448475835000571280108248, 19.72688266700824000832796484883, 20.82051059620421311246845505373, 21.78324744398313604462678584099, 22.57110253584090368740849054920, 23.623131447176632877821740620664, 24.64023664418296311097850598989, 25.76355187257771523346809428056, 26.40840113129493789678876316651, 26.71563378114026727763274797620

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