Properties

Label 1-199-199.64-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.740 + 0.672i$
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  + (−0.995 + 0.0950i)2-s + (0.235 + 0.971i)3-s + (0.981 − 0.189i)4-s + (−0.142 + 0.989i)5-s + (−0.327 − 0.945i)6-s + (0.723 + 0.690i)7-s + (−0.959 + 0.281i)8-s + (−0.888 + 0.458i)9-s + (0.0475 − 0.998i)10-s + (0.841 + 0.540i)11-s + (0.415 + 0.909i)12-s + (−0.995 + 0.0950i)13-s + (−0.786 − 0.618i)14-s + (−0.995 + 0.0950i)15-s + (0.928 − 0.371i)16-s + (0.841 + 0.540i)17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)2-s + (0.235 + 0.971i)3-s + (0.981 − 0.189i)4-s + (−0.142 + 0.989i)5-s + (−0.327 − 0.945i)6-s + (0.723 + 0.690i)7-s + (−0.959 + 0.281i)8-s + (−0.888 + 0.458i)9-s + (0.0475 − 0.998i)10-s + (0.841 + 0.540i)11-s + (0.415 + 0.909i)12-s + (−0.995 + 0.0950i)13-s + (−0.786 − 0.618i)14-s + (−0.995 + 0.0950i)15-s + (0.928 − 0.371i)16-s + (0.841 + 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.740 + 0.672i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.740 + 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2878628548 + 0.7450598816i\)
\(L(\frac12)\) \(\approx\) \(0.2878628548 + 0.7450598816i\)
\(L(1)\) \(\approx\) \(0.6041167297 + 0.4842994165i\)
\(L(1)\) \(\approx\) \(0.6041167297 + 0.4842994165i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.995 + 0.0950i)T \)
3 \( 1 + (0.235 + 0.971i)T \)
5 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (0.723 + 0.690i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (-0.995 + 0.0950i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.981 + 0.189i)T \)
29 \( 1 + (0.580 - 0.814i)T \)
31 \( 1 + (0.235 - 0.971i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.888 + 0.458i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.723 + 0.690i)T \)
53 \( 1 + (-0.995 - 0.0950i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (-0.786 + 0.618i)T \)
73 \( 1 + (0.981 + 0.189i)T \)
79 \( 1 + (0.928 - 0.371i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (0.580 - 0.814i)T \)
97 \( 1 + (0.235 - 0.971i)T \)
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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

−26.91768060886936383091045586882, −25.282929952872168732586422386233, −24.87960918469577899933132250109, −24.07056959720352797196632354935, −23.20507363407839410075275404405, −21.35799349984426282022220924271, −20.50311784249979169018573157107, −19.71593138735663232516618883777, −19.05184567443783191968374996693, −17.84879409787050782265975858543, −16.99869224050840389440246262076, −16.53155623416794934516025922704, −14.81800059437768902666964854302, −13.918040732400637556051208988045, −12.418194239095651439984968413685, −11.96597967741715402793643928695, −10.71576777534919564489334883800, −9.31402184538619137910682794323, −8.42711603131789028210678214154, −7.64734191647526343463403588722, −6.68136648524422305677546309806, −5.18333161787770764550495987425, −3.37739800524506565927794638184, −1.72666904527200255621277465747, −0.86391337544622682394943153681, 2.08184317236244072194792697793, 3.10284768835757761258478646323, 4.71304998757962143001621071616, 6.12213301874899644148773971341, 7.355143372876671014551916972352, 8.423008602744377133296575258127, 9.47280555432584445479314111190, 10.25329393510376841911976809020, 11.30450643197323378544195544350, 11.98946917972043102068418624535, 14.25993274427867428024999806955, 15.11590523298179474391317769494, 15.33177504625737442114052514003, 17.05284308561414684349074315265, 17.39927961993148020622694325637, 18.85378122283308760254855038553, 19.43234140058620731606674747674, 20.56482062214875477229938623473, 21.5558834015227341513406429475, 22.22552212738115772475805880256, 23.58560730089863426689899750202, 24.930270981939013690998261800155, 25.58214923220080298377216295941, 26.50700527133028648378569020758, 27.32089604040489735934250348966

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