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Properties

Label 1-197-197.116-r0-0-0
Degree $1$
Conductor $197$
Sign $-0.956 + 0.292i$
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  + (0.345 + 0.938i)2-s + (0.838 + 0.545i)3-s + (−0.761 + 0.648i)4-s + (−0.0320 + 0.999i)5-s + (−0.222 + 0.974i)6-s + (−0.345 + 0.938i)7-s + (−0.871 − 0.490i)8-s + (0.404 + 0.914i)9-s + (−0.949 + 0.315i)10-s + (−0.718 − 0.695i)11-s + (−0.991 + 0.127i)12-s + (0.572 − 0.820i)13-s − 14-s + (−0.572 + 0.820i)15-s + (0.159 − 0.987i)16-s + (0.462 − 0.886i)17-s + ⋯
L(s)  = 1  + (0.345 + 0.938i)2-s + (0.838 + 0.545i)3-s + (−0.761 + 0.648i)4-s + (−0.0320 + 0.999i)5-s + (−0.222 + 0.974i)6-s + (−0.345 + 0.938i)7-s + (−0.871 − 0.490i)8-s + (0.404 + 0.914i)9-s + (−0.949 + 0.315i)10-s + (−0.718 − 0.695i)11-s + (−0.991 + 0.127i)12-s + (0.572 − 0.820i)13-s − 14-s + (−0.572 + 0.820i)15-s + (0.159 − 0.987i)16-s + (0.462 − 0.886i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(197\)
Sign: $-0.956 + 0.292i$
Analytic conductor: \(0.914864\)
Root analytic conductor: \(0.914864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{197} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 197,\ (0:\ ),\ -0.956 + 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2245556909 + 1.503779541i\)
\(L(\frac12)\) \(\approx\) \(0.2245556909 + 1.503779541i\)
\(L(1)\) \(\approx\) \(0.8217385197 + 1.105847603i\)
\(L(1)\) \(\approx\) \(0.8217385197 + 1.105847603i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 \)
good2 \( 1 + (0.345 + 0.938i)T \)
3 \( 1 + (0.838 + 0.545i)T \)
5 \( 1 + (-0.0320 + 0.999i)T \)
7 \( 1 + (-0.345 + 0.938i)T \)
11 \( 1 + (-0.718 - 0.695i)T \)
13 \( 1 + (0.572 - 0.820i)T \)
17 \( 1 + (0.462 - 0.886i)T \)
19 \( 1 + (0.623 - 0.781i)T \)
23 \( 1 + (-0.0960 + 0.995i)T \)
29 \( 1 + (0.991 - 0.127i)T \)
31 \( 1 + (-0.518 + 0.855i)T \)
37 \( 1 + (0.159 + 0.987i)T \)
41 \( 1 + (-0.462 + 0.886i)T \)
43 \( 1 + (0.718 + 0.695i)T \)
47 \( 1 + (0.284 - 0.958i)T \)
53 \( 1 + (0.967 - 0.253i)T \)
59 \( 1 + (-0.949 - 0.315i)T \)
61 \( 1 + (-0.838 + 0.545i)T \)
67 \( 1 + (-0.284 + 0.958i)T \)
71 \( 1 + (-0.404 - 0.914i)T \)
73 \( 1 + (-0.159 - 0.987i)T \)
79 \( 1 + (-0.0320 - 0.999i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.518 - 0.855i)T \)
97 \( 1 + (0.801 - 0.598i)T \)
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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

−26.500000795273388667727819555257, −25.71112892581977060715148171224, −24.34804853732314607756544009300, −23.6529118586494259003332190970, −22.96021097280345186766263782793, −21.30737487429232847555595714554, −20.62686634225869496833904720418, −20.08971311640318697273557053694, −19.1445327708593415362401273752, −18.25016862261845062630194692050, −17.02825169468710008581598103975, −15.74329272841572038836284473386, −14.359809449310338245627660028249, −13.650247565019993341082678266044, −12.72234273407358340797414273098, −12.20443147728651405074306853059, −10.56559011811174223975482648824, −9.62488661661002457544151673314, −8.63707982269685656920569886575, −7.5606391171249048021949639365, −5.995490873195160343979304862593, −4.41346245941826691734604224075, −3.67648294291443661158547648429, −2.12565777793110934504185534781, −1.04822530338567415332523458497, 2.99889710643431856845764084975, 3.171108055070101355914667228970, 5.00361890571434491353218689183, 5.95574210590604209920644096415, 7.31671080567908072004111889905, 8.21832306059692238799128305469, 9.23260267429104904606776294661, 10.28775183778673126241987704205, 11.69830676903602091987357985106, 13.27074547141913023996423988191, 13.85760723304449957644826026723, 14.990990703325176467916141147263, 15.62900858104388929948700337638, 16.20614958689173287043239646088, 18.00961033054314259498317565698, 18.513747025775274425346087626947, 19.6489886596332858164827003269, 21.15398858568821535832186215965, 21.794956297904112118907817825693, 22.57987385008596527092993149011, 23.59733772646849531448675082424, 24.96518605424386967276440012894, 25.44961403305716234853166397451, 26.26176171902485938806586407904, 27.03833672837735865150226475043

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