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Properties

Label 1-197-197.15-r0-0-0
Degree $1$
Conductor $197$
Sign $-0.634 + 0.772i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.718 + 0.695i)2-s + (−0.518 + 0.855i)3-s + (0.0320 − 0.999i)4-s + (−0.801 − 0.598i)5-s + (−0.222 − 0.974i)6-s + (0.718 + 0.695i)7-s + (0.672 + 0.740i)8-s + (−0.462 − 0.886i)9-s + (0.991 − 0.127i)10-s + (0.949 − 0.315i)11-s + (0.838 + 0.545i)12-s + (−0.926 + 0.375i)13-s − 14-s + (0.926 − 0.375i)15-s + (−0.997 − 0.0640i)16-s + (0.981 + 0.191i)17-s + ⋯
L(s)  = 1  + (−0.718 + 0.695i)2-s + (−0.518 + 0.855i)3-s + (0.0320 − 0.999i)4-s + (−0.801 − 0.598i)5-s + (−0.222 − 0.974i)6-s + (0.718 + 0.695i)7-s + (0.672 + 0.740i)8-s + (−0.462 − 0.886i)9-s + (0.991 − 0.127i)10-s + (0.949 − 0.315i)11-s + (0.838 + 0.545i)12-s + (−0.926 + 0.375i)13-s − 14-s + (0.926 − 0.375i)15-s + (−0.997 − 0.0640i)16-s + (0.981 + 0.191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(197\)
Sign: $-0.634 + 0.772i$
Analytic conductor: \(0.914864\)
Root analytic conductor: \(0.914864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{197} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 197,\ (0:\ ),\ -0.634 + 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2381855960 + 0.5041026766i\)
\(L(\frac12)\) \(\approx\) \(0.2381855960 + 0.5041026766i\)
\(L(1)\) \(\approx\) \(0.4911442632 + 0.3429404971i\)
\(L(1)\) \(\approx\) \(0.4911442632 + 0.3429404971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 \)
good2 \( 1 + (-0.718 + 0.695i)T \)
3 \( 1 + (-0.518 + 0.855i)T \)
5 \( 1 + (-0.801 - 0.598i)T \)
7 \( 1 + (0.718 + 0.695i)T \)
11 \( 1 + (0.949 - 0.315i)T \)
13 \( 1 + (-0.926 + 0.375i)T \)
17 \( 1 + (0.981 + 0.191i)T \)
19 \( 1 + (0.623 + 0.781i)T \)
23 \( 1 + (-0.345 + 0.938i)T \)
29 \( 1 + (-0.838 - 0.545i)T \)
31 \( 1 + (0.0960 + 0.995i)T \)
37 \( 1 + (-0.997 + 0.0640i)T \)
41 \( 1 + (-0.981 - 0.191i)T \)
43 \( 1 + (-0.949 + 0.315i)T \)
47 \( 1 + (0.871 - 0.490i)T \)
53 \( 1 + (0.404 + 0.914i)T \)
59 \( 1 + (0.991 + 0.127i)T \)
61 \( 1 + (0.518 + 0.855i)T \)
67 \( 1 + (-0.871 + 0.490i)T \)
71 \( 1 + (0.462 + 0.886i)T \)
73 \( 1 + (0.997 - 0.0640i)T \)
79 \( 1 + (-0.801 + 0.598i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.0960 - 0.995i)T \)
97 \( 1 + (0.967 - 0.253i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.93526303272827797567858986839, −25.86341775130235224571614978473, −24.6833492144870607968275336377, −23.83139300262137479663664148547, −22.58158751229809534235291613261, −22.16802434079506428214048878061, −20.466146766964738955412540459869, −19.81040017494649889083200462569, −18.934545244206280012136662240159, −18.094963709290344562482672234501, −17.211736789460490198287366080593, −16.50763942118250822364079757469, −14.833041548144527878595007326838, −13.771984876413204166250669777938, −12.375377025224888642299198294, −11.759811525850750620573652786234, −10.96513183254780100159444171697, −9.946009997145137159325085779025, −8.32324946761775866226331792448, −7.42853511578479313221948154079, −6.885335175062271853107780535149, −4.85317349670601890559606736512, −3.49272873338717465713675219774, −2.062996333714700465011346042646, −0.65865493026009975290356848772, 1.3823628985583740741278725184, 3.76387728526238025397402683902, 5.0394408199819604641268006153, 5.70871344044835023318374403144, 7.25885618508347112872378333885, 8.408343918110355400978858667687, 9.22336281615696445879649080324, 10.20338669423072793672363102432, 11.71275140219723018687768663317, 11.90196353628376104691959816027, 14.23579517080882729023860791738, 14.971060306793257086206577769179, 15.83298473879300948615939316148, 16.754661993642431142760737019114, 17.289749881644225504669635911796, 18.58455047703497745956212445083, 19.55425443884494075109811406283, 20.54783102698420398779148578866, 21.64034669193779667960804171900, 22.726603983472496330555350584317, 23.72518161937054229458887286075, 24.47276885911241220944353813764, 25.367818347622712042657237388713, 26.83056085029572500944872976071, 27.21458066060036223643334668716

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