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Properties

Label 1-199-199.29-r0-0-0
Degree $1$
Conductor $199$
Sign $0.841 - 0.541i$
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  + (−0.987 + 0.158i)2-s + (−0.605 + 0.795i)3-s + (0.950 − 0.312i)4-s + (0.235 − 0.971i)5-s + (0.472 − 0.881i)6-s + (0.296 + 0.954i)7-s + (−0.888 + 0.458i)8-s + (−0.266 − 0.963i)9-s + (−0.0792 + 0.996i)10-s + (0.415 − 0.909i)11-s + (−0.327 + 0.945i)12-s + (0.356 − 0.934i)13-s + (−0.444 − 0.895i)14-s + (0.630 + 0.776i)15-s + (0.805 − 0.592i)16-s + (−0.995 + 0.0950i)17-s + ⋯
L(s)  = 1  + (−0.987 + 0.158i)2-s + (−0.605 + 0.795i)3-s + (0.950 − 0.312i)4-s + (0.235 − 0.971i)5-s + (0.472 − 0.881i)6-s + (0.296 + 0.954i)7-s + (−0.888 + 0.458i)8-s + (−0.266 − 0.963i)9-s + (−0.0792 + 0.996i)10-s + (0.415 − 0.909i)11-s + (−0.327 + 0.945i)12-s + (0.356 − 0.934i)13-s + (−0.444 − 0.895i)14-s + (0.630 + 0.776i)15-s + (0.805 − 0.592i)16-s + (−0.995 + 0.0950i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.841 - 0.541i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.841 - 0.541i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5642609255 - 0.1658221102i\)
\(L(\frac12)\) \(\approx\) \(0.5642609255 - 0.1658221102i\)
\(L(1)\) \(\approx\) \(0.6109517016 + 0.001054377498i\)
\(L(1)\) \(\approx\) \(0.6109517016 + 0.001054377498i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.987 + 0.158i)T \)
3 \( 1 + (-0.605 + 0.795i)T \)
5 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (0.296 + 0.954i)T \)
11 \( 1 + (0.415 - 0.909i)T \)
13 \( 1 + (0.356 - 0.934i)T \)
17 \( 1 + (-0.995 + 0.0950i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.204 - 0.978i)T \)
29 \( 1 + (-0.0158 - 0.999i)T \)
31 \( 1 + (0.991 - 0.126i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (0.967 + 0.251i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.678 - 0.734i)T \)
53 \( 1 + (0.630 - 0.776i)T \)
59 \( 1 + (0.928 - 0.371i)T \)
61 \( 1 + (-0.959 - 0.281i)T \)
67 \( 1 + (0.723 + 0.690i)T \)
71 \( 1 + (-0.444 + 0.895i)T \)
73 \( 1 + (-0.745 + 0.666i)T \)
79 \( 1 + (-0.916 - 0.400i)T \)
83 \( 1 + (-0.327 - 0.945i)T \)
89 \( 1 + (0.873 + 0.486i)T \)
97 \( 1 + (-0.386 + 0.922i)T \)
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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

−27.06033045265485563984873550573, −25.997360284274388160228898019610, −25.4443410680366637311016627410, −24.15900265904347732201396290663, −23.408868818140021053866183957233, −22.3097791369387928860504091942, −21.25183448935132859092886512146, −19.91763438959847473492083152203, −19.265997215495794852918371633806, −18.21301988838050387709924932662, −17.53633297534172693859832367400, −16.94318381187883686976401850277, −15.60778945916621973861115340986, −14.31682105416851368830334694719, −13.25479468358688406451749856995, −11.90183158424214372218582086394, −11.06303066415311678637000776390, −10.411326584038187159837669945265, −9.10065809148105773674265252869, −7.613369319441144397340525424985, −6.936611823103835783798917582052, −6.26568212762348767361329869443, −4.23713774648374042132993225716, −2.412261759231771622275421597169, −1.4187982743225436134951945130, 0.71684566925862800171335246944, 2.46421355946035615550843702644, 4.3110829154046587356101536524, 5.73936164290114899694821924026, 6.180436008675120996619898213987, 8.34468471830888685320766251590, 8.74860763938015493638402826101, 9.82298497911670088169537879286, 10.92330344608838699559464566851, 11.75795861533895117904917618133, 12.82183408758437254687982218860, 14.65417175497882694224381925792, 15.63789689932612997431451205517, 16.26814209504626343794278096648, 17.26400311325845572432288255180, 17.88112817779046700054886537115, 19.09378939024290210579035346123, 20.25794794553397012601122404255, 21.07849692626760471609922313308, 21.78787139550983162138011151197, 23.12241248206018923014462297661, 24.50253461277253908840635491986, 24.77321714962371963871298379541, 26.07381882114821466244481216157, 27.0891026831120701955966418845

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