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Properties

Label 1-199-199.45-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.922 - 0.386i$
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  + (0.950 − 0.312i)2-s + (−0.266 − 0.963i)3-s + (0.805 − 0.592i)4-s + (−0.888 − 0.458i)5-s + (−0.553 − 0.832i)6-s + (−0.823 + 0.567i)7-s + (0.580 − 0.814i)8-s + (−0.857 + 0.513i)9-s + (−0.987 − 0.158i)10-s + (−0.654 − 0.755i)11-s + (−0.786 − 0.618i)12-s + (−0.745 − 0.666i)13-s + (−0.605 + 0.795i)14-s + (−0.204 + 0.978i)15-s + (0.296 − 0.954i)16-s + (0.981 − 0.189i)17-s + ⋯
L(s)  = 1  + (0.950 − 0.312i)2-s + (−0.266 − 0.963i)3-s + (0.805 − 0.592i)4-s + (−0.888 − 0.458i)5-s + (−0.553 − 0.832i)6-s + (−0.823 + 0.567i)7-s + (0.580 − 0.814i)8-s + (−0.857 + 0.513i)9-s + (−0.987 − 0.158i)10-s + (−0.654 − 0.755i)11-s + (−0.786 − 0.618i)12-s + (−0.745 − 0.666i)13-s + (−0.605 + 0.795i)14-s + (−0.204 + 0.978i)15-s + (0.296 − 0.954i)16-s + (0.981 − 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.922 - 0.386i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.922 - 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2365652155 - 1.177299545i\)
\(L(\frac12)\) \(\approx\) \(0.2365652155 - 1.177299545i\)
\(L(1)\) \(\approx\) \(0.8945350934 - 0.8076845871i\)
\(L(1)\) \(\approx\) \(0.8945350934 - 0.8076845871i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.950 - 0.312i)T \)
3 \( 1 + (-0.266 - 0.963i)T \)
5 \( 1 + (-0.888 - 0.458i)T \)
7 \( 1 + (-0.823 + 0.567i)T \)
11 \( 1 + (-0.654 - 0.755i)T \)
13 \( 1 + (-0.745 - 0.666i)T \)
17 \( 1 + (0.981 - 0.189i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.916 + 0.400i)T \)
29 \( 1 + (-0.999 + 0.0317i)T \)
31 \( 1 + (0.967 - 0.251i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.873 + 0.486i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.0792 - 0.996i)T \)
53 \( 1 + (-0.204 - 0.978i)T \)
59 \( 1 + (0.723 - 0.690i)T \)
61 \( 1 + (0.841 + 0.540i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (-0.605 - 0.795i)T \)
73 \( 1 + (0.110 - 0.993i)T \)
79 \( 1 + (0.678 + 0.734i)T \)
83 \( 1 + (-0.786 + 0.618i)T \)
89 \( 1 + (0.527 + 0.849i)T \)
97 \( 1 + (-0.701 - 0.712i)T \)
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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

−27.074840508715150335095095816, −26.232629186560133311192070935, −25.753339359877505752440960014441, −24.13649790824104950739317948617, −23.25107962960497419090575181450, −22.68042291620797092708432643922, −21.98953512933025107814546927919, −20.72857952688545706697433482682, −20.12547777094062627484739262025, −18.93848387489213352850910894237, −17.2607948464099595229963442749, −16.32225497272969387301675776547, −15.76477184860237760914533998340, −14.77863325258680841637557916474, −14.01900244710098127147625886804, −12.45033353596855881333475994647, −11.82885048715495829219550436206, −10.5726510488370380284864665915, −9.79205984892566323875535197590, −7.93034988168686374003697263530, −7.04721229226693846623368275178, −5.82433000522090933932555546791, −4.552306996885037116538577077896, −3.78211656940516847417246692477, −2.78287044929574143705457794042, 0.68171199597909241924454745870, 2.54053743982569113363037204695, 3.44648051576053261745759409198, 5.19112967885928430396867966213, 5.85240879006108513497026531583, 7.23663062561726668605557026995, 8.063372526515372251602227468523, 9.74959748701989153275711219384, 11.22472540142146040282923825303, 11.98841599815005198901539042116, 12.74413242279260062991630990252, 13.44302703720491715316875786245, 14.72394267203818069261589004554, 15.88497458813117111064610689726, 16.49528140220238820220031918198, 18.15037853815331494627850342381, 19.27876303408340170708663143846, 19.60473332546216071163428670681, 20.77231963588917486298470210618, 22.08934149221825770105566393199, 22.79479327784092047687832449814, 23.64829399823996977500061989240, 24.414552952982287059443042934390, 25.06780736343510497655044325694, 26.34041388772768174678127067066

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