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Properties

Label 1-199-199.140-r0-0-0
Degree $1$
Conductor $199$
Sign $0.339 - 0.940i$
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  + (−0.786 + 0.618i)2-s + (−0.995 + 0.0950i)3-s + (0.235 − 0.971i)4-s + (0.841 − 0.540i)5-s + (0.723 − 0.690i)6-s + (0.580 − 0.814i)7-s + (0.415 + 0.909i)8-s + (0.981 − 0.189i)9-s + (−0.327 + 0.945i)10-s + (−0.654 − 0.755i)11-s + (−0.142 + 0.989i)12-s + (−0.786 + 0.618i)13-s + (0.0475 + 0.998i)14-s + (−0.786 + 0.618i)15-s + (−0.888 − 0.458i)16-s + (−0.654 − 0.755i)17-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (−0.995 + 0.0950i)3-s + (0.235 − 0.971i)4-s + (0.841 − 0.540i)5-s + (0.723 − 0.690i)6-s + (0.580 − 0.814i)7-s + (0.415 + 0.909i)8-s + (0.981 − 0.189i)9-s + (−0.327 + 0.945i)10-s + (−0.654 − 0.755i)11-s + (−0.142 + 0.989i)12-s + (−0.786 + 0.618i)13-s + (0.0475 + 0.998i)14-s + (−0.786 + 0.618i)15-s + (−0.888 − 0.458i)16-s + (−0.654 − 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.339 - 0.940i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.339 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4443376611 - 0.3120683283i\)
\(L(\frac12)\) \(\approx\) \(0.4443376611 - 0.3120683283i\)
\(L(1)\) \(\approx\) \(0.5900204007 - 0.06589072734i\)
\(L(1)\) \(\approx\) \(0.5900204007 - 0.06589072734i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.786 + 0.618i)T \)
3 \( 1 + (-0.995 + 0.0950i)T \)
5 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (0.580 - 0.814i)T \)
11 \( 1 + (-0.654 - 0.755i)T \)
13 \( 1 + (-0.786 + 0.618i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.235 + 0.971i)T \)
29 \( 1 + (0.928 - 0.371i)T \)
31 \( 1 + (-0.995 - 0.0950i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.981 - 0.189i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.580 - 0.814i)T \)
53 \( 1 + (-0.786 - 0.618i)T \)
59 \( 1 + (-0.959 - 0.281i)T \)
61 \( 1 + (0.841 + 0.540i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.0475 - 0.998i)T \)
73 \( 1 + (0.235 + 0.971i)T \)
79 \( 1 + (-0.888 - 0.458i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
89 \( 1 + (0.928 - 0.371i)T \)
97 \( 1 + (-0.995 - 0.0950i)T \)
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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

−27.31158304596444989790898385240, −26.3769849709831612000870248774, −25.23010920673085386403501983770, −24.57302568525724188433021909475, −23.05539367280283081638275361200, −22.067256511141710132432984670735, −21.51977094651309596549825680987, −20.59947377027966006934837297738, −19.15145498246942764078315154551, −18.1929834781514557293246311178, −17.7691839751639876501130765395, −16.99807356975890384837342220007, −15.70597316751146127419590929582, −14.597385381052877279784561805553, −12.6976324758398224138062622293, −12.510157012946185692158954090776, −10.98210109228776882267201602923, −10.48025685356751746728975632886, −9.49143565972763595264578199985, −8.09992695643891104696835787457, −6.949687838580653785606233526207, −5.79480012365237187388611539395, −4.5750783850736914855101791143, −2.577342659020526080171292620742, −1.712182036040370997417643484642, 0.60519241908129667174835268079, 1.98727290591449816343794701171, 4.68763771219082827044483687921, 5.29897028882638524460323577625, 6.53521682593926873703035905722, 7.41823738497472186305933730338, 8.825779841091591015680715405809, 9.845224730522547402821418958436, 10.75023869731377528471309598632, 11.5965596674883515682626985669, 13.24829721113061935538226560921, 14.05309282053396464033209806442, 15.51390673523673926956471381076, 16.43019244243331061542757141172, 17.23671855884724874843987097079, 17.67777541286807917400133788988, 18.733019130609565845419664028054, 19.98654465850211212746409488063, 21.1006471283951010193773056813, 21.94761446946459845758799857294, 23.40365466928398697292592299003, 24.02145881216338412689784518126, 24.63631662072617820048120278337, 25.937263612214304928353301106937, 26.89214495055720461225592272545

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