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Properties

Label 1-197-197.9-r0-0-0
Degree $1$
Conductor $197$
Sign $-0.985 + 0.167i$
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  + (0.572 − 0.820i)2-s + (−0.284 + 0.958i)3-s + (−0.345 − 0.938i)4-s + (−0.718 + 0.695i)5-s + (0.623 + 0.781i)6-s + (−0.572 − 0.820i)7-s + (−0.967 − 0.253i)8-s + (−0.838 − 0.545i)9-s + (0.159 + 0.987i)10-s + (−0.926 − 0.375i)11-s + (0.997 − 0.0640i)12-s + (0.462 + 0.886i)13-s − 14-s + (−0.462 − 0.886i)15-s + (−0.761 + 0.648i)16-s + (−0.518 − 0.855i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.820i)2-s + (−0.284 + 0.958i)3-s + (−0.345 − 0.938i)4-s + (−0.718 + 0.695i)5-s + (0.623 + 0.781i)6-s + (−0.572 − 0.820i)7-s + (−0.967 − 0.253i)8-s + (−0.838 − 0.545i)9-s + (0.159 + 0.987i)10-s + (−0.926 − 0.375i)11-s + (0.997 − 0.0640i)12-s + (0.462 + 0.886i)13-s − 14-s + (−0.462 − 0.886i)15-s + (−0.761 + 0.648i)16-s + (−0.518 − 0.855i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(197\)
Sign: $-0.985 + 0.167i$
Analytic conductor: \(0.914864\)
Root analytic conductor: \(0.914864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{197} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 197,\ (0:\ ),\ -0.985 + 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01277117612 - 0.1516500270i\)
\(L(\frac12)\) \(\approx\) \(0.01277117612 - 0.1516500270i\)
\(L(1)\) \(\approx\) \(0.6515118911 - 0.1758504109i\)
\(L(1)\) \(\approx\) \(0.6515118911 - 0.1758504109i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 \)
good2 \( 1 + (0.572 - 0.820i)T \)
3 \( 1 + (-0.284 + 0.958i)T \)
5 \( 1 + (-0.718 + 0.695i)T \)
7 \( 1 + (-0.572 - 0.820i)T \)
11 \( 1 + (-0.926 - 0.375i)T \)
13 \( 1 + (0.462 + 0.886i)T \)
17 \( 1 + (-0.518 - 0.855i)T \)
19 \( 1 + (-0.900 + 0.433i)T \)
23 \( 1 + (-0.672 - 0.740i)T \)
29 \( 1 + (-0.997 + 0.0640i)T \)
31 \( 1 + (-0.871 + 0.490i)T \)
37 \( 1 + (-0.761 - 0.648i)T \)
41 \( 1 + (0.518 + 0.855i)T \)
43 \( 1 + (0.926 + 0.375i)T \)
47 \( 1 + (0.801 - 0.598i)T \)
53 \( 1 + (0.991 - 0.127i)T \)
59 \( 1 + (0.159 - 0.987i)T \)
61 \( 1 + (0.284 + 0.958i)T \)
67 \( 1 + (-0.801 + 0.598i)T \)
71 \( 1 + (0.838 + 0.545i)T \)
73 \( 1 + (0.761 + 0.648i)T \)
79 \( 1 + (-0.718 - 0.695i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.871 - 0.490i)T \)
97 \( 1 + (-0.949 + 0.315i)T \)
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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

−27.65716439425893435053349919183, −25.855263365951570862176798941332, −25.607286456403334138322275817621, −24.27506341205353963220565413742, −23.91744957649708743706827821364, −22.95183163285558161412009848217, −22.19990275149802025163994072512, −20.88815483646421236141046940817, −19.74368126599542881355104136078, −18.672855192549122002397862176461, −17.731475988700538194355843334656, −16.80579827718383655950548329904, −15.62622489751075222671871756386, −15.19695861754601875890306402226, −13.47384434707913730639310596860, −12.76784696832601507462411520377, −12.3226450112704621439082335683, −11.01389419760718692278677948342, −8.96113449362683181613589824281, −8.142581941212408440120836040253, −7.29996181592524554008329346769, −5.96982416864431105761119054655, −5.319313291588984937382862398052, −3.80582172963413895322446505234, −2.33917923406584277551950502286, 0.09344500195828425847321881070, 2.59717571794529674509326910739, 3.75828262763426990251472662763, 4.32909186225183989607622040836, 5.7919337467074658502457159068, 6.94982097533970556794089526460, 8.72641912306994818678134894742, 9.96627689709126432995320518271, 10.801429220316500386192300470846, 11.300595445518571618265329919225, 12.585435157281755221093681253556, 13.845795508684253450953142634257, 14.64479904830572023469718346966, 15.79926352033715770240446437411, 16.419160926240001843097098084884, 18.09885637515412389796006549271, 19.00625590734940510145889793678, 20.01930321411670952413407299924, 20.80610187126621002286597384060, 21.74594415100611265787318061271, 22.68875061513777909225522934340, 23.23283997008803469626658657141, 24.00958169453300122888695279069, 26.0330257766334672901091663615, 26.59759393854086123826011369522

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