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Properties

Label 1-283-283.59-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.677 + 0.735i$
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 + (−0.338 + 0.940i)3-s + (0.991 − 0.133i)4-s + (0.937 + 0.348i)5-s + (0.274 − 0.961i)6-s + (0.188 − 0.982i)7-s + (−0.979 + 0.199i)8-s + (−0.770 − 0.637i)9-s + (−0.958 − 0.285i)10-s + (−0.610 + 0.791i)11-s + (−0.210 + 0.977i)12-s + (−0.970 + 0.242i)13-s + (−0.122 + 0.992i)14-s + (−0.645 + 0.763i)15-s + (0.964 − 0.264i)16-s + (−0.798 + 0.602i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0667i)2-s + (−0.338 + 0.940i)3-s + (0.991 − 0.133i)4-s + (0.937 + 0.348i)5-s + (0.274 − 0.961i)6-s + (0.188 − 0.982i)7-s + (−0.979 + 0.199i)8-s + (−0.770 − 0.637i)9-s + (−0.958 − 0.285i)10-s + (−0.610 + 0.791i)11-s + (−0.210 + 0.977i)12-s + (−0.970 + 0.242i)13-s + (−0.122 + 0.992i)14-s + (−0.645 + 0.763i)15-s + (0.964 − 0.264i)16-s + (−0.798 + 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.677 + 0.735i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.677 + 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2342701676 + 0.5343044230i\)
\(L(\frac12)\) \(\approx\) \(0.2342701676 + 0.5343044230i\)
\(L(1)\) \(\approx\) \(0.5532164330 + 0.2940840458i\)
\(L(1)\) \(\approx\) \(0.5532164330 + 0.2940840458i\)

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 \( 1 + (-0.338 + 0.940i)T \)
5 \( 1 + (0.937 + 0.348i)T \)
7 \( 1 + (0.188 - 0.982i)T \)
11 \( 1 + (-0.610 + 0.791i)T \)
13 \( 1 + (-0.970 + 0.242i)T \)
17 \( 1 + (-0.798 + 0.602i)T \)
19 \( 1 + (0.695 + 0.718i)T \)
23 \( 1 + (-0.999 + 0.0222i)T \)
29 \( 1 + (0.100 + 0.994i)T \)
31 \( 1 + (-0.679 + 0.734i)T \)
37 \( 1 + (0.999 - 0.0445i)T \)
41 \( 1 + (0.317 + 0.948i)T \)
43 \( 1 + (-0.420 + 0.907i)T \)
47 \( 1 + (0.274 + 0.961i)T \)
53 \( 1 + (0.964 + 0.264i)T \)
59 \( 1 + (0.144 + 0.989i)T \)
61 \( 1 + (0.231 - 0.972i)T \)
67 \( 1 + (-0.741 + 0.670i)T \)
71 \( 1 + (0.991 + 0.133i)T \)
73 \( 1 + (-0.679 - 0.734i)T \)
79 \( 1 + (-0.420 - 0.907i)T \)
83 \( 1 + (-0.871 - 0.490i)T \)
89 \( 1 + (-0.575 - 0.818i)T \)
97 \( 1 + (0.519 + 0.854i)T \)
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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

−25.17106320283906165434575291792, −24.36153481376601586431476304981, −24.1702764192048587949206504779, −22.25310820083415901386838883045, −21.6686462388117578383981658209, −20.46984460179998786976543624569, −19.60909952762395787550599199069, −18.49404977480530654827563181249, −18.089253038389802932845052126716, −17.28732345919208995856241199629, −16.37365635737450643481933254892, −15.32293130024252944206002208528, −13.92705187971032107750173291522, −12.96726633590589071264992563012, −11.97399465124734790981121069954, −11.23049549263560761056501138597, −9.96463000976951477378146831193, −8.99242779469903976539710030078, −8.13973143237749048700338692816, −7.08050771972273698601124876697, −5.91979018010725214060320763007, −5.31647731139166896865554246199, −2.54186781687640684660165514565, −2.19215962308699745896316450914, −0.530498242812252304603709935467, 1.64048543961312565633606899808, 2.93622624370713866459160182077, 4.48306490130347707836925946975, 5.67843609198932678570190287388, 6.768225778601778392843072835445, 7.76549640963373476807486917291, 9.17653148118791836684767370783, 10.05174148069865911627589378565, 10.395316994708026218221498019002, 11.40445472263199839664024610565, 12.70410346248625512010241127687, 14.288922419034386819547561403591, 14.879395145322224729997414575861, 16.14483247269287925956733472057, 16.81243617766476134725256658771, 17.74454059204343114670761409062, 18.10392535183293499038680525713, 19.833625492440373966847344943538, 20.31122313508394884903256681056, 21.317823979385695124618405538577, 22.048518320331813805237172052524, 23.23406875108074138191522695269, 24.22173416125319475006146312154, 25.38989704946069578797509814291, 26.24870892147216927241325072841

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