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Properties

Label 1-197-197.16-r0-0-0
Degree $1$
Conductor $197$
Sign $0.501 - 0.864i$
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  + (0.991 + 0.127i)2-s + (−0.345 − 0.938i)3-s + (0.967 + 0.253i)4-s + (0.404 − 0.914i)5-s + (−0.222 − 0.974i)6-s + (0.991 − 0.127i)7-s + (0.926 + 0.375i)8-s + (−0.761 + 0.648i)9-s + (0.518 − 0.855i)10-s + (−0.838 − 0.545i)11-s + (−0.0960 − 0.995i)12-s + (−0.997 − 0.0640i)13-s + 14-s + (−0.997 − 0.0640i)15-s + (0.871 + 0.490i)16-s + (0.0320 + 0.999i)17-s + ⋯
L(s)  = 1  + (0.991 + 0.127i)2-s + (−0.345 − 0.938i)3-s + (0.967 + 0.253i)4-s + (0.404 − 0.914i)5-s + (−0.222 − 0.974i)6-s + (0.991 − 0.127i)7-s + (0.926 + 0.375i)8-s + (−0.761 + 0.648i)9-s + (0.518 − 0.855i)10-s + (−0.838 − 0.545i)11-s + (−0.0960 − 0.995i)12-s + (−0.997 − 0.0640i)13-s + 14-s + (−0.997 − 0.0640i)15-s + (0.871 + 0.490i)16-s + (0.0320 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(197\)
Sign: $0.501 - 0.864i$
Analytic conductor: \(0.914864\)
Root analytic conductor: \(0.914864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{197} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 197,\ (0:\ ),\ 0.501 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.781267778 - 1.025835882i\)
\(L(\frac12)\) \(\approx\) \(1.781267778 - 1.025835882i\)
\(L(1)\) \(\approx\) \(1.686327828 - 0.5744136810i\)
\(L(1)\) \(\approx\) \(1.686327828 - 0.5744136810i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 \)
good2 \( 1 + (0.991 + 0.127i)T \)
3 \( 1 + (-0.345 - 0.938i)T \)
5 \( 1 + (0.404 - 0.914i)T \)
7 \( 1 + (0.991 - 0.127i)T \)
11 \( 1 + (-0.838 - 0.545i)T \)
13 \( 1 + (-0.997 - 0.0640i)T \)
17 \( 1 + (0.0320 + 0.999i)T \)
19 \( 1 + (0.623 + 0.781i)T \)
23 \( 1 + (-0.949 + 0.315i)T \)
29 \( 1 + (-0.0960 - 0.995i)T \)
31 \( 1 + (0.718 - 0.695i)T \)
37 \( 1 + (0.871 - 0.490i)T \)
41 \( 1 + (0.0320 + 0.999i)T \)
43 \( 1 + (-0.838 - 0.545i)T \)
47 \( 1 + (-0.572 + 0.820i)T \)
53 \( 1 + (-0.981 + 0.191i)T \)
59 \( 1 + (0.518 + 0.855i)T \)
61 \( 1 + (-0.345 + 0.938i)T \)
67 \( 1 + (-0.572 + 0.820i)T \)
71 \( 1 + (-0.761 + 0.648i)T \)
73 \( 1 + (0.871 - 0.490i)T \)
79 \( 1 + (0.404 + 0.914i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.718 + 0.695i)T \)
97 \( 1 + (-0.462 - 0.886i)T \)
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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

−27.03512663486210022706018729423, −26.219179021192171961366045784067, −25.168686558036727588329127680026, −24.03537725641262842269678850374, −23.11321040792133365266302178854, −22.16977986998422991881340244378, −21.68924744541600402096786526828, −20.74234480473644512073044801066, −19.973026766787967874637133009205, −18.300207557839756551363396085478, −17.49389786883584266403713675831, −16.15343798556734761224568140473, −15.23470856974895766140476499017, −14.54169609372735969068118131508, −13.76512512623324243332076034766, −12.16594413933404021889091016523, −11.34919124751925379978355815355, −10.47320499014079281415517980819, −9.64562976877349119543426070265, −7.705092048301327538538756484220, −6.60355384711516731825028996236, −5.161754776014493359402691999501, −4.813057294308966874776030160757, −3.18234896931748862767969244329, −2.23105417811711303197090931395, 1.440586983266844669879859963705, 2.46809679408225701311578774161, 4.376192699402974343138906076, 5.42753294106620763076152639460, 6.06317563950110969074787165741, 7.79029760111131338104302247831, 8.07406755079756709850177482624, 10.17820878095704368311788222964, 11.478672092869694086487936987313, 12.203473864719155362078760760357, 13.15382222230183617335660767352, 13.85183751598878083156867381233, 14.86939538465845690889443648804, 16.29159730044337976528691994654, 17.09000261727732009585634919524, 17.92359884968774088703589157542, 19.340373136247816862247115984225, 20.33462760812173590318376113313, 21.20287558158479989772660295884, 22.06344962063986699280570346061, 23.333600809235746472683220618740, 24.11761709153943541296414289109, 24.463853923329392822090023912796, 25.35086210163456824762425037620, 26.643610192063716073569605172287

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