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Properties

Label 1-283-283.70-r0-0-0
Degree $1$
Conductor $283$
Sign $0.926 - 0.376i$
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  + (0.100 − 0.994i)2-s + (−0.711 + 0.703i)3-s + (−0.979 − 0.199i)4-s + (−0.871 − 0.490i)5-s + (0.628 + 0.777i)6-s + (−0.999 + 0.0222i)7-s + (−0.296 + 0.955i)8-s + (0.0111 − 0.999i)9-s + (−0.575 + 0.818i)10-s + (0.662 + 0.749i)11-s + (0.836 − 0.547i)12-s + (−0.987 + 0.155i)13-s + (−0.0779 + 0.996i)14-s + (0.964 − 0.264i)15-s + (0.920 + 0.390i)16-s + (0.902 + 0.431i)17-s + ⋯
L(s)  = 1  + (0.100 − 0.994i)2-s + (−0.711 + 0.703i)3-s + (−0.979 − 0.199i)4-s + (−0.871 − 0.490i)5-s + (0.628 + 0.777i)6-s + (−0.999 + 0.0222i)7-s + (−0.296 + 0.955i)8-s + (0.0111 − 0.999i)9-s + (−0.575 + 0.818i)10-s + (0.662 + 0.749i)11-s + (0.836 − 0.547i)12-s + (−0.987 + 0.155i)13-s + (−0.0779 + 0.996i)14-s + (0.964 − 0.264i)15-s + (0.920 + 0.390i)16-s + (0.902 + 0.431i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.926 - 0.376i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.926 - 0.376i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5670228184 - 0.1109354704i\)
\(L(\frac12)\) \(\approx\) \(0.5670228184 - 0.1109354704i\)
\(L(1)\) \(\approx\) \(0.6001477504 - 0.1765635777i\)
\(L(1)\) \(\approx\) \(0.6001477504 - 0.1765635777i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.100 - 0.994i)T \)
3 \( 1 + (-0.711 + 0.703i)T \)
5 \( 1 + (-0.871 - 0.490i)T \)
7 \( 1 + (-0.999 + 0.0222i)T \)
11 \( 1 + (0.662 + 0.749i)T \)
13 \( 1 + (-0.987 + 0.155i)T \)
17 \( 1 + (0.902 + 0.431i)T \)
19 \( 1 + (0.359 - 0.933i)T \)
23 \( 1 + (-0.848 - 0.528i)T \)
29 \( 1 + (0.593 + 0.805i)T \)
31 \( 1 + (0.188 - 0.982i)T \)
37 \( 1 + (0.441 + 0.897i)T \)
41 \( 1 + (0.975 + 0.220i)T \)
43 \( 1 + (0.991 + 0.133i)T \)
47 \( 1 + (0.628 - 0.777i)T \)
53 \( 1 + (0.920 - 0.390i)T \)
59 \( 1 + (-0.460 + 0.887i)T \)
61 \( 1 + (-0.420 + 0.907i)T \)
67 \( 1 + (-0.892 + 0.451i)T \)
71 \( 1 + (-0.979 + 0.199i)T \)
73 \( 1 + (0.188 + 0.982i)T \)
79 \( 1 + (0.991 - 0.133i)T \)
83 \( 1 + (-0.970 + 0.242i)T \)
89 \( 1 + (-0.380 - 0.924i)T \)
97 \( 1 + (0.882 - 0.470i)T \)
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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

−25.35641350027485401393723219044, −24.70945618956133271052179589538, −23.783587093818392574306627906079, −22.94872177510905731523019918905, −22.5258393001165804525518739575, −21.66777209294735224155579972876, −19.57322217262435370240079514064, −19.168373241356332881395480781615, −18.28261537270420026015800243840, −17.23066763875715197669495168401, −16.31232028088272503398602217007, −15.86015257534756031906540784533, −14.42594055414056754532062199537, −13.79260473769937277797543425903, −12.33257764697699067865759282934, −12.088048122215943475973295918638, −10.52416407959154100366850001227, −9.418169481464196447618206663050, −7.90979563613082279638836697978, −7.38128045491558577264867792658, −6.34520138045366547221413550266, −5.64184871911031557135866132662, −4.176549985707684801605695096000, −3.04636385075339376250390469935, −0.633818450345244454125296951753, 0.85211174223864199154994109991, 2.83620571889389094054343981095, 4.020094621658784820456671400671, 4.60663783414557586555726728337, 5.83638322597620578175651174494, 7.26750061506109521939007973433, 8.86288081144588287872117994294, 9.67575426743859119270489310608, 10.3791092282923757764101494637, 11.725015516727246946037152705755, 12.14293942644660549246550585565, 12.927882462902949043533180532981, 14.500247856383604636571933642897, 15.36263476063780093027210833738, 16.50431697290383552337235164435, 17.16901878308810686771612901305, 18.34209096918877432653734475110, 19.58430368564297186230534055869, 19.929208364430899802332985265801, 20.9916055701501134826225770930, 22.12035562555814365638992220937, 22.51110684022143085657560951181, 23.40267646936687756718600945391, 24.21773646404472880765017592633, 25.89881211069599073298776110053

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