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Properties

Label 1-199-199.196-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.999 + 0.0337i$
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  + (−0.975 − 0.220i)2-s + (−0.999 − 0.0317i)3-s + (0.902 + 0.429i)4-s + (−0.327 − 0.945i)5-s + (0.967 + 0.251i)6-s + (−0.204 − 0.978i)7-s + (−0.786 − 0.618i)8-s + (0.997 + 0.0634i)9-s + (0.110 + 0.993i)10-s + (−0.959 + 0.281i)11-s + (−0.888 − 0.458i)12-s + (0.678 − 0.734i)13-s + (−0.0158 + 0.999i)14-s + (0.296 + 0.954i)15-s + (0.630 + 0.776i)16-s + (0.723 + 0.690i)17-s + ⋯
L(s)  = 1  + (−0.975 − 0.220i)2-s + (−0.999 − 0.0317i)3-s + (0.902 + 0.429i)4-s + (−0.327 − 0.945i)5-s + (0.967 + 0.251i)6-s + (−0.204 − 0.978i)7-s + (−0.786 − 0.618i)8-s + (0.997 + 0.0634i)9-s + (0.110 + 0.993i)10-s + (−0.959 + 0.281i)11-s + (−0.888 − 0.458i)12-s + (0.678 − 0.734i)13-s + (−0.0158 + 0.999i)14-s + (0.296 + 0.954i)15-s + (0.630 + 0.776i)16-s + (0.723 + 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.999 + 0.0337i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.999 + 0.0337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004150304819 - 0.2458673492i\)
\(L(\frac12)\) \(\approx\) \(0.004150304819 - 0.2458673492i\)
\(L(1)\) \(\approx\) \(0.3578347362 - 0.1948206803i\)
\(L(1)\) \(\approx\) \(0.3578347362 - 0.1948206803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.975 - 0.220i)T \)
3 \( 1 + (-0.999 - 0.0317i)T \)
5 \( 1 + (-0.327 - 0.945i)T \)
7 \( 1 + (-0.204 - 0.978i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (0.678 - 0.734i)T \)
17 \( 1 + (0.723 + 0.690i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.823 - 0.567i)T \)
29 \( 1 + (-0.605 + 0.795i)T \)
31 \( 1 + (0.472 - 0.881i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.553 + 0.832i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.745 + 0.666i)T \)
53 \( 1 + (0.296 - 0.954i)T \)
59 \( 1 + (-0.995 + 0.0950i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (-0.0158 - 0.999i)T \)
73 \( 1 + (-0.0792 + 0.996i)T \)
79 \( 1 + (0.356 - 0.934i)T \)
83 \( 1 + (-0.888 + 0.458i)T \)
89 \( 1 + (-0.386 - 0.922i)T \)
97 \( 1 + (0.527 + 0.849i)T \)
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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

−27.47408262444880382749647745417, −26.53445660724838296443773871863, −25.73209037459254148801118080203, −24.65146290821542315912664194336, −23.572690161491778134602425152622, −22.88973989989425705078078970902, −21.64639324307685787466437443549, −20.88211652700518242456325881960, −19.18682013855518351184787686990, −18.43781028594143169110752053816, −18.24311534320014439774495436189, −16.796327187513335262869811380292, −15.84719295107866141357996023485, −15.40198850073416836149133592130, −13.933196548399218473226676612615, −12.09066604294985725367058308043, −11.59115931922732734153676443829, −10.49627605168660804038359602392, −9.75898635449392891809820859556, −8.289912413300120914698975797, −7.22711598808222647885995186169, −6.18146258867923123959499577005, −5.42884110001742355627067600656, −3.33873019831781157034405388530, −1.85668881657766909731157903225, 0.30048729616973110284638005706, 1.48587611307197834554919873693, 3.56032886791071250886845702095, 4.934635854451378066900142012070, 6.22325331669134810334492956305, 7.48908592090380887429049714561, 8.24783671535595122203850971359, 9.77741717249713785754286421731, 10.526806151972309081366909840692, 11.42089058340185365925946932517, 12.61938104760643461024064757570, 13.16977180548619043012776846223, 15.40476496728971683132142601034, 16.169868799056190666414136365120, 16.88558722651650587981293761264, 17.69666780695913191346133033227, 18.58895956858346580544106843414, 19.82968908287449128728755601715, 20.560271398689022379469964848913, 21.418113649770923586325690164636, 22.846669966312254016451948344996, 23.76656507061798709240224753399, 24.38041777910384969795749025846, 25.7831610152998164172742382454, 26.54699844116390180840662938577

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