Properties

Label 1-199-199.61-r0-0-0
Degree $1$
Conductor $199$
Sign $0.492 + 0.870i$
Analytic cond. $0.924152$
Root an. cond. $0.924152$
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  + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (0.415 − 0.909i)5-s + (0.841 + 0.540i)6-s + (−0.654 + 0.755i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.142 + 0.989i)10-s + (−0.142 + 0.989i)11-s + (−0.959 − 0.281i)12-s + (−0.959 + 0.281i)13-s + (0.415 − 0.909i)14-s + (−0.959 + 0.281i)15-s + (0.415 − 0.909i)16-s + (−0.142 + 0.989i)17-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (0.415 − 0.909i)5-s + (0.841 + 0.540i)6-s + (−0.654 + 0.755i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.142 + 0.989i)10-s + (−0.142 + 0.989i)11-s + (−0.959 − 0.281i)12-s + (−0.959 + 0.281i)13-s + (0.415 − 0.909i)14-s + (−0.959 + 0.281i)15-s + (0.415 − 0.909i)16-s + (−0.142 + 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.492 + 0.870i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3801367998 + 0.2217619259i\)
\(L(\frac12)\) \(\approx\) \(0.3801367998 + 0.2217619259i\)
\(L(1)\) \(\approx\) \(0.5208043153 + 0.02996662909i\)
\(L(1)\) \(\approx\) \(0.5208043153 + 0.02996662909i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 + (-0.654 - 0.755i)T \)
5 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (-0.654 + 0.755i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + T \)
23 \( 1 + (0.841 + 0.540i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.654 + 0.755i)T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−26.7176529110260314185826769570, −26.386767671210340328971674959, −25.18712893304343426315030850889, −23.986190048489477762839585182202, −22.4798906848467128071822691744, −22.183418875306681202432175933305, −20.96884614691015143379072649885, −20.1181728770170257723100557229, −18.92906390252332392638133165908, −18.12236672619202594519064612144, −17.07389286770149024196178498448, −16.45329330141298618217610818553, −15.478520709591958628535393110411, −14.26074157391704590618966451968, −12.851220780793514920687019720167, −11.40351178158851712185189429028, −10.87118872825686355311944951481, −9.85286492573144567996879497610, −9.32076250619150415172842166489, −7.49160440477018639610979649891, −6.67306132059503732925575175066, −5.48395288806697400494034481185, −3.6121334466032882780461506937, −2.75189540806088129001763641829, −0.51590701472270335131932304444, 1.39622439644498700151079087260, 2.41779293428529460489673595662, 5.03563582838150768214965337201, 5.85613729544737841795968065828, 6.9638937444859738869032893634, 7.93999365098812147042486659091, 9.24051228083135986566322523957, 9.89190124105995873389043579338, 11.36800113589657521762695980960, 12.3818528339989743192909757698, 12.99587561713387173038736888008, 14.67333078014294164054132334713, 15.85881088988905317447652916088, 16.75395960058236397467691677329, 17.483424994487546585185037235802, 18.25981718086302455871147364732, 19.35339676676056386098742223161, 19.98769533950577101296534710315, 21.33117335901865769876489754841, 22.45124488679077891301640910068, 23.67908489620310632322131323473, 24.45426762207728085039319293095, 25.17477034405825304219574366383, 25.8582243358417360719004190916, 27.2877538454539413890137658664

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