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Properties

Label 1-283-283.102-r1-0-0
Degree $1$
Conductor $283$
Sign $0.810 + 0.586i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
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  + (0.420 − 0.907i)2-s + (−0.593 + 0.805i)3-s + (−0.645 − 0.763i)4-s + (0.296 − 0.955i)5-s + (0.480 + 0.876i)6-s + (−0.824 − 0.565i)7-s + (−0.964 + 0.264i)8-s + (−0.296 − 0.955i)9-s + (−0.741 − 0.670i)10-s + (−0.645 + 0.763i)11-s + (0.997 − 0.0667i)12-s + (0.480 + 0.876i)13-s + (−0.860 + 0.509i)14-s + (0.593 + 0.805i)15-s + (−0.166 + 0.986i)16-s + (−0.860 − 0.509i)17-s + ⋯
L(s)  = 1  + (0.420 − 0.907i)2-s + (−0.593 + 0.805i)3-s + (−0.645 − 0.763i)4-s + (0.296 − 0.955i)5-s + (0.480 + 0.876i)6-s + (−0.824 − 0.565i)7-s + (−0.964 + 0.264i)8-s + (−0.296 − 0.955i)9-s + (−0.741 − 0.670i)10-s + (−0.645 + 0.763i)11-s + (0.997 − 0.0667i)12-s + (0.480 + 0.876i)13-s + (−0.860 + 0.509i)14-s + (0.593 + 0.805i)15-s + (−0.166 + 0.986i)16-s + (−0.860 − 0.509i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.810 + 0.586i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.810 + 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6165451723 + 0.1995825539i\)
\(L(\frac12)\) \(\approx\) \(0.6165451723 + 0.1995825539i\)
\(L(1)\) \(\approx\) \(0.7312135743 - 0.3037053206i\)
\(L(1)\) \(\approx\) \(0.7312135743 - 0.3037053206i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.420 - 0.907i)T \)
3 \( 1 + (-0.593 + 0.805i)T \)
5 \( 1 + (0.296 - 0.955i)T \)
7 \( 1 + (-0.824 - 0.565i)T \)
11 \( 1 + (-0.645 + 0.763i)T \)
13 \( 1 + (0.480 + 0.876i)T \)
17 \( 1 + (-0.860 - 0.509i)T \)
19 \( 1 + (-0.480 - 0.876i)T \)
23 \( 1 + (0.784 + 0.619i)T \)
29 \( 1 + (0.991 - 0.133i)T \)
31 \( 1 + (-0.920 + 0.390i)T \)
37 \( 1 + (-0.231 - 0.972i)T \)
41 \( 1 + (0.964 + 0.264i)T \)
43 \( 1 + (0.892 - 0.451i)T \)
47 \( 1 + (-0.480 + 0.876i)T \)
53 \( 1 + (0.166 + 0.986i)T \)
59 \( 1 + (0.359 + 0.933i)T \)
61 \( 1 + (-0.741 + 0.670i)T \)
67 \( 1 + (0.997 + 0.0667i)T \)
71 \( 1 + (-0.645 + 0.763i)T \)
73 \( 1 + (0.920 + 0.390i)T \)
79 \( 1 + (0.892 + 0.451i)T \)
83 \( 1 + (-0.944 - 0.328i)T \)
89 \( 1 + (-0.892 + 0.451i)T \)
97 \( 1 + (0.784 + 0.619i)T \)
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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

−25.24838224811363154941180259223, −24.390352640204796683327962719153, −23.35864408313586605083609852234, −22.69539091289064407179512850614, −22.1103960136793981406463020745, −21.17256202113313131731641449694, −19.36741043323035204748945062965, −18.5302336983714785344329824878, −18.02686584730431404508753776635, −16.98071101138747100046173277451, −16.01455966883491920330459904177, −15.17055076589983003537568006027, −14.06331740869279079049314020090, −13.08349006860176949056089373708, −12.65066837271397776238039806906, −11.23288404443588930359602259101, −10.28504197923039769202422226044, −8.686011781784985668235361068212, −7.804985892006009010211595455250, −6.545959069940712963616850615808, −6.174404128423513385080416754589, −5.25823107064725394706403901881, −3.45023006118423108173949030569, −2.46703059495417834118310878907, −0.2252692851891287526608278576, 0.97073251009038062780925278555, 2.58669070367625908815305922521, 4.07389402247594310934928560240, 4.63369334387564896341123981477, 5.66520744861941358230546607599, 6.84282519961099556925834063069, 9.04353480970111617604810952997, 9.36715635711623980328777217730, 10.48553001674044283823992080416, 11.25155792447349757070328015131, 12.433214518888170616721070005306, 13.09174763692221064293913678743, 14.03954547043131256658764730054, 15.4848226026326483949610765970, 16.11305605237915242462004233565, 17.28868536279109009014249770654, 17.99900711506408883814697995938, 19.48762203650162268032353330371, 20.177613340575630874646002312671, 21.07647003252712511366011215822, 21.5790778014562343539261820802, 22.73481920582559835119229717088, 23.35338027575947537841818955347, 24.08564261058354000964049240493, 25.63457068139778420751032765760

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