↑

Properties

Label 1-283-283.24-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

Related objects

Downloads

Learn more

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: $\chi_{283} (24, \cdot )$
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 \( 1 + (0.519 - 0.854i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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