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Properties

Label 1-283-283.175-r0-0-0
Degree $1$
Conductor $283$
Sign $0.695 - 0.718i$
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  + (−0.997 + 0.0667i)2-s + (−0.645 − 0.763i)3-s + (0.991 − 0.133i)4-s + (−0.166 − 0.986i)5-s + (0.695 + 0.718i)6-s + (−0.944 + 0.328i)7-s + (−0.979 + 0.199i)8-s + (−0.166 + 0.986i)9-s + (0.231 + 0.972i)10-s + (0.991 + 0.133i)11-s + (−0.741 − 0.670i)12-s + (0.695 + 0.718i)13-s + (0.920 − 0.390i)14-s + (−0.645 + 0.763i)15-s + (0.964 − 0.264i)16-s + (0.920 + 0.390i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0667i)2-s + (−0.645 − 0.763i)3-s + (0.991 − 0.133i)4-s + (−0.166 − 0.986i)5-s + (0.695 + 0.718i)6-s + (−0.944 + 0.328i)7-s + (−0.979 + 0.199i)8-s + (−0.166 + 0.986i)9-s + (0.231 + 0.972i)10-s + (0.991 + 0.133i)11-s + (−0.741 − 0.670i)12-s + (0.695 + 0.718i)13-s + (0.920 − 0.390i)14-s + (−0.645 + 0.763i)15-s + (0.964 − 0.264i)16-s + (0.920 + 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5496516535 - 0.2327900604i\)
\(L(\frac12)\) \(\approx\) \(0.5496516535 - 0.2327900604i\)
\(L(1)\) \(\approx\) \(0.5612221052 - 0.1522035270i\)
\(L(1)\) \(\approx\) \(0.5612221052 - 0.1522035270i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.997 + 0.0667i)T \)
3 \( 1 + (-0.645 - 0.763i)T \)
5 \( 1 + (-0.166 - 0.986i)T \)
7 \( 1 + (-0.944 + 0.328i)T \)
11 \( 1 + (0.991 + 0.133i)T \)
13 \( 1 + (0.695 + 0.718i)T \)
17 \( 1 + (0.920 + 0.390i)T \)
19 \( 1 + (0.695 + 0.718i)T \)
23 \( 1 + (0.480 - 0.876i)T \)
29 \( 1 + (0.100 + 0.994i)T \)
31 \( 1 + (-0.296 - 0.955i)T \)
37 \( 1 + (-0.538 - 0.842i)T \)
41 \( 1 + (-0.979 - 0.199i)T \)
43 \( 1 + (-0.420 + 0.907i)T \)
47 \( 1 + (0.695 - 0.718i)T \)
53 \( 1 + (0.964 + 0.264i)T \)
59 \( 1 + (0.784 - 0.619i)T \)
61 \( 1 + (0.231 - 0.972i)T \)
67 \( 1 + (-0.741 + 0.670i)T \)
71 \( 1 + (0.991 + 0.133i)T \)
73 \( 1 + (-0.296 + 0.955i)T \)
79 \( 1 + (-0.420 - 0.907i)T \)
83 \( 1 + (0.860 - 0.509i)T \)
89 \( 1 + (-0.420 + 0.907i)T \)
97 \( 1 + (0.480 - 0.876i)T \)
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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

−25.74816425257861133081090472538, −25.368692402780312734269094687390, −23.74766336222527806220964000518, −22.799625391780761182839017289604, −22.16964302454028250843953819710, −21.1293869479349541345527217843, −20.06566455255591371797342469615, −19.271562288465607516760847194538, −18.305473908933864778965581089708, −17.43236310957869209269432367155, −16.60798230834617926967359072366, −15.69843292765344514106858308990, −15.109793792618689696962732023069, −13.72143663171506745460429995591, −12.06092734270799421121990142324, −11.408126329430242064554454825245, −10.40664610374614596265412849077, −9.84032605326304931905936123800, −8.87016407056397473156119332038, −7.320141697177983237057584304763, −6.56885208398128340893760007119, −5.63524096261326829667776207701, −3.59712856916443255270709839802, −3.11802970658685455801831339313, −0.92020357566795303356318520971, 0.9015863559890348490324461976, 1.86083716377512870212495286668, 3.63057453053908012773659128969, 5.45560024779494078070562386575, 6.31248021557257639066263592598, 7.20061556810725040567815743112, 8.40956375934008650153215644587, 9.17097280084798791597963213901, 10.23534979701042206992652554714, 11.581431316734628943000486087255, 12.16950851592045767426620094220, 12.97142133043801805056539673936, 14.380111132637280320755457886168, 15.889679355328387741279671277574, 16.64620394683087816282058617549, 16.94377101330616615464183952233, 18.314154574792213443609529417709, 18.94611640360766992472226091891, 19.723444388564255066184717120267, 20.62441173025694941718615570119, 21.829643003025064128165056431636, 23.02342881861316703230607852254, 23.83239753997046982121988159231, 24.85837759095494411984061591194, 25.197332843121745537842210259651

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