↑

Properties

Label 1-197-197.28-r0-0-0
Degree $1$
Conductor $197$
Sign $0.660 - 0.750i$
Analytic cond. $0.914864$
Root an. cond. $0.914864$
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  + (0.0320 − 0.999i)2-s + (−0.462 − 0.886i)3-s + (−0.997 − 0.0640i)4-s + (0.284 + 0.958i)5-s + (−0.900 + 0.433i)6-s + (0.0320 + 0.999i)7-s + (−0.0960 + 0.995i)8-s + (−0.572 + 0.820i)9-s + (0.967 − 0.253i)10-s + (0.801 − 0.598i)11-s + (0.404 + 0.914i)12-s + (0.718 − 0.695i)13-s + 14-s + (0.718 − 0.695i)15-s + (0.991 + 0.127i)16-s + (0.926 + 0.375i)17-s + ⋯
L(s)  = 1  + (0.0320 − 0.999i)2-s + (−0.462 − 0.886i)3-s + (−0.997 − 0.0640i)4-s + (0.284 + 0.958i)5-s + (−0.900 + 0.433i)6-s + (0.0320 + 0.999i)7-s + (−0.0960 + 0.995i)8-s + (−0.572 + 0.820i)9-s + (0.967 − 0.253i)10-s + (0.801 − 0.598i)11-s + (0.404 + 0.914i)12-s + (0.718 − 0.695i)13-s + 14-s + (0.718 − 0.695i)15-s + (0.991 + 0.127i)16-s + (0.926 + 0.375i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(197\)
Sign: $0.660 - 0.750i$
Analytic conductor: \(0.914864\)
Root analytic conductor: \(0.914864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{197} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 197,\ (0:\ ),\ 0.660 - 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8890495973 - 0.4016588132i\)
\(L(\frac12)\) \(\approx\) \(0.8890495973 - 0.4016588132i\)
\(L(1)\) \(\approx\) \(0.8442711670 - 0.3896656625i\)
\(L(1)\) \(\approx\) \(0.8442711670 - 0.3896656625i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 \)
good2 \( 1 + (0.0320 - 0.999i)T \)
3 \( 1 + (-0.462 - 0.886i)T \)
5 \( 1 + (0.284 + 0.958i)T \)
7 \( 1 + (0.0320 + 0.999i)T \)
11 \( 1 + (0.801 - 0.598i)T \)
13 \( 1 + (0.718 - 0.695i)T \)
17 \( 1 + (0.926 + 0.375i)T \)
19 \( 1 + (-0.222 + 0.974i)T \)
23 \( 1 + (-0.761 - 0.648i)T \)
29 \( 1 + (0.404 + 0.914i)T \)
31 \( 1 + (-0.981 + 0.191i)T \)
37 \( 1 + (0.991 - 0.127i)T \)
41 \( 1 + (0.926 + 0.375i)T \)
43 \( 1 + (0.801 - 0.598i)T \)
47 \( 1 + (0.518 - 0.855i)T \)
53 \( 1 + (-0.672 + 0.740i)T \)
59 \( 1 + (0.967 + 0.253i)T \)
61 \( 1 + (-0.462 + 0.886i)T \)
67 \( 1 + (0.518 - 0.855i)T \)
71 \( 1 + (-0.572 + 0.820i)T \)
73 \( 1 + (0.991 - 0.127i)T \)
79 \( 1 + (0.284 - 0.958i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.981 - 0.191i)T \)
97 \( 1 + (0.871 - 0.490i)T \)
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

−27.12498400916810615503676917085, −26.00363989290478034254788391426, −25.415765361358180719875992966333, −24.01568555390522807380417757128, −23.48588107540855838815979298654, −22.55020382790910699247057638700, −21.489926450350919582048507436817, −20.63918232481553415691156577995, −19.55122132072689597514451942085, −17.82170467210797474206803996846, −17.20478062995537971335254281016, −16.44034977764485067035734689639, −15.81055069339175413949316872176, −14.49318073224710993370973545240, −13.71685733346355738510377987881, −12.50534358695066108912850640942, −11.2386480461827839873293417054, −9.67760581820440426184613572975, −9.34664914582635514965875505381, −7.97300401805665490643418528158, −6.64575384171271740017351816749, −5.61879892725388724574297459311, −4.439130570199904081086561222045, −3.94305893948634452820900898247, −0.97600658396190395382341266341, 1.369733099575852393062444296370, 2.527513162192543115369056370202, 3.62184168321553630547306833377, 5.651382640161067514383554742643, 6.12405416688445415947484047490, 7.84882191524593601557337808184, 8.858124402993704961047721773437, 10.307957812777488856045903696419, 11.11394702676158260043262086686, 12.07382123226385117181529967266, 12.79409280182024139805665015256, 14.06971173954731918840829422158, 14.66062090413133755640268927528, 16.47916924466828466010473252207, 17.73856842191761987205880521384, 18.46774894801106630458423016851, 18.90193121541480147087256204131, 19.950860851307170308646254698307, 21.36094975565569613206222526730, 22.11835033128739419648125097042, 22.79494784474315077025479587501, 23.722978730477839599700785164362, 25.064571763169506097547667765466, 25.7444468837094130300587525047, 27.23507315758488358072920596541

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