Properties

Label 1-283-283.103-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.944 + 0.327i$
Analytic cond. $1.31424$
Root an. cond. $1.31424$
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  + (−0.296 + 0.955i)2-s + (−0.970 + 0.242i)3-s + (−0.824 − 0.565i)4-s + (−0.848 + 0.528i)5-s + (0.0556 − 0.998i)6-s + (0.556 + 0.830i)7-s + (0.784 − 0.619i)8-s + (0.882 − 0.470i)9-s + (−0.253 − 0.967i)10-s + (0.902 + 0.431i)11-s + (0.937 + 0.348i)12-s + (0.836 + 0.547i)13-s + (−0.958 + 0.285i)14-s + (0.695 − 0.718i)15-s + (0.359 + 0.933i)16-s + (0.726 − 0.687i)17-s + ⋯
L(s)  = 1  + (−0.296 + 0.955i)2-s + (−0.970 + 0.242i)3-s + (−0.824 − 0.565i)4-s + (−0.848 + 0.528i)5-s + (0.0556 − 0.998i)6-s + (0.556 + 0.830i)7-s + (0.784 − 0.619i)8-s + (0.882 − 0.470i)9-s + (−0.253 − 0.967i)10-s + (0.902 + 0.431i)11-s + (0.937 + 0.348i)12-s + (0.836 + 0.547i)13-s + (−0.958 + 0.285i)14-s + (0.695 − 0.718i)15-s + (0.359 + 0.933i)16-s + (0.726 − 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.944 + 0.327i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.944 + 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1058336547 + 0.6285532209i\)
\(L(\frac12)\) \(\approx\) \(0.1058336547 + 0.6285532209i\)
\(L(1)\) \(\approx\) \(0.4556459910 + 0.4525776077i\)
\(L(1)\) \(\approx\) \(0.4556459910 + 0.4525776077i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.296 + 0.955i)T \)
3 \( 1 + (-0.970 + 0.242i)T \)
5 \( 1 + (-0.848 + 0.528i)T \)
7 \( 1 + (0.556 + 0.830i)T \)
11 \( 1 + (0.902 + 0.431i)T \)
13 \( 1 + (0.836 + 0.547i)T \)
17 \( 1 + (0.726 - 0.687i)T \)
19 \( 1 + (-0.892 + 0.451i)T \)
23 \( 1 + (0.811 + 0.584i)T \)
29 \( 1 + (-0.944 + 0.328i)T \)
31 \( 1 + (-0.460 - 0.887i)T \)
37 \( 1 + (0.317 + 0.948i)T \)
41 \( 1 + (-0.929 + 0.369i)T \)
43 \( 1 + (0.920 + 0.390i)T \)
47 \( 1 + (0.0556 + 0.998i)T \)
53 \( 1 + (0.359 - 0.933i)T \)
59 \( 1 + (-0.610 - 0.791i)T \)
61 \( 1 + (0.964 - 0.264i)T \)
67 \( 1 + (-0.166 + 0.986i)T \)
71 \( 1 + (-0.824 + 0.565i)T \)
73 \( 1 + (-0.460 + 0.887i)T \)
79 \( 1 + (0.920 - 0.390i)T \)
83 \( 1 + (-0.210 - 0.977i)T \)
89 \( 1 + (-0.122 - 0.992i)T \)
97 \( 1 + (-0.911 + 0.410i)T \)
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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

−25.08886228963480089600740984489, −23.856101896878732627197794917703, −23.32325345543180084197985285331, −22.54629408441664036719751469513, −21.41496321614818353185696517847, −20.64075416714164326462421989982, −19.62487832471181311135745594160, −18.945077134799403001819820734659, −17.8751784385128400004830239101, −16.8877907044155553234939605412, −16.58897037388491289000151714334, −14.99787704464042075467203904188, −13.60358541100564185531174170686, −12.72107659110962412087684922371, −11.95475656075724733915768044981, −10.90695070510740125613024121088, −10.67731752454825913999441545367, −8.994098874070661405244894186932, −8.085887939497247994455804817433, −7.05951977141258245513241046959, −5.49118423622997710897271847239, −4.30774069551622617315713043528, −3.66094675391287551976015006773, −1.5250142161045562306592696062, −0.66270871886063125246987387855, 1.39336709555527073917338749741, 3.770485062309491538400000506725, 4.70005922282967471367175446149, 5.82298554294183928397159379918, 6.689925178579720347760283293463, 7.62471506118541304355626643478, 8.81423076761127770828697094408, 9.78139830830146177823256679773, 11.11855618213907565432843583430, 11.68797612441390743792924774099, 12.87756550652906285645005940349, 14.49572697818176037742454373619, 15.02302669997027626026782957167, 15.92949455221780794232700953433, 16.73468907068817994536721179986, 17.63108104531184337135284011452, 18.7192852564442421433392609661, 18.91260001032013085997932631444, 20.6580580878748612742681849086, 21.88258398936465201992283066255, 22.60280837762180823158041702359, 23.37909060008806404660349772482, 23.99671422838706892929421196839, 25.09059265617391898749621133228, 25.92392699902207457171075097198

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